This section covers:
 Introduction to Vectors
 Vector Operations
 Applications of Vectors
 Dot Product and Angle Between Two Vectors
 3D Vectors – Vectors in Space (including Cross Product)
 Parametric Form of the Equation of a Line in Space
 More Practice
Introduction to Vectors
A vector (also called a direction vector) is just is something that has both magnitude (length, or size) and direction. It’s different than a regular number, since it really has two components to it. We see vectors represented by arrows, so we can remember that we need to get a length of a vector (the magnitude), as well as the direction (which way it’s pointing).
We use vectors in mathematics, engineering, and physics, since many times we need to know both the size of something and which way it’s going. For example, with an airplane, we can use a vector to measure the speed of the plane (the “size”) and the direction it’s flying.
Geometric Vectors are directed line segments in the \(xy\)plane, and, as an example, the vector from a point \(A\) (initial point) to a point \(B\) (terminal point) can be represented by \(\overrightarrow{{AB}}\) .
So, for example, if \(A\) is \((2,7)\) and \(B\) is \((3,8)\), the vector is second point minus first point, or \(\displaystyle \left\langle {{{x}_{2}}} \right.{{x}_{1}},\left. {{{y}_{2}}{{y}_{1}}} \right\rangle \), or \(\left\langle {32} \right.,\left. {87} \right\rangle =\left\langle {5,\left. 1 \right\rangle } \right.\). The “\(x\)” part of the vector (–5) is called the \(x\)component, and the ”\(y\)” part (1) is called the \(y\)component. This \(\left\langle {x,} \right.\left. y \right\rangle \) form is called component form.
Note also that vectors can also be written in the form \(ai+bj\) (called the linear combination of the unit vectors \(i\) and \(j\)), so this vector can also be written as \(5i+1j\), or \(5i+j\).
The magnitude of the vector, written \(\left\ {AB} \right\\) is the distance between the two points (like the hypotenuse of a right triangle), or \(\sqrt{{{{{\left( {{{x}_{2}}{{x}_{1}}} \right)}}^{2}}+{{{\left( {{{y}_{2}}{{y}_{1}}} \right)}}^{2}}}}\), or with the new vector \(\left\langle {x,} \right.\left. y \right\rangle \), it’s just \(\sqrt{{{{x}^{2}}+{{y}^{2}}}}\). So for our points \(A\) and \(B\) above, \(\left\ {AB} \right\=\sqrt{{{{{\left( {5} \right)}}^{2}}+{{1}^{2}}}}=\sqrt{{26}}\).
Now looking at this vector visually, do you see how we can use the slope of the line of the vector (from the initial point to the terminal point) to get the direction of the vector? Pretty cool! So we can just use \(\displaystyle {{\tan }^{{1}}}\left( {\frac{y}{x}} \right)\) (second part of vector over first part of vector) to get the angle measurement of the vector’s direction. Remembering from the Polar Coordinates, Equations and Graphs section though, we have to be careful which quadrant the vector terminates in (“pretending” that the vector’s initial point is at the origin) to know how many degrees we should add to that tangent value when we use a calculator:
Here is all this visually. Note that we had to add 180° to the angle measurement we got from the calculator (–11.3°) since the vector would terminate in the 2^{nd} quadrant if we were to start at \((0,0)\). So, we get 168.7°, which is the angle measurement from the positive x axis going counterclockwise.
168.7° from the positive \(x\)axis can also be described as 11.3° North of West (11.3° N of W, or W11.3°W), since the closest axis to the angle is the negative \(x\)axis (west) and we are going a little north of that:
(We saw a similar concept of this when we were working with bearings here in the Law of Sines and Cosines, and Areas of Triangles section).
Note that a vector that has a magnitude of 0 (and thus no direction) is called a zero vector. So, hypothetically, the vector \(\overrightarrow{{AA}}\) would be a zero vector.
A unit vector is a vector with magnitude 1; in some applications, it’s easier to work with unit vectors. To find the unit vector that is associated with a vector (has same direction, but magnitude of 1), use the following formula: \(\displaystyle u=\frac{v}{{\left\ v \right\}}\) (just divide each component of a vector by its magnitude to get its unit vector). We’ll see some problems below.
Vector Operations
Adding and Subtracting Vectors
There are a couple of ways to add and subtract vectors. When we add vectors, geometrically, we just put the beginning point (initial point) of the second vector at the end point (terminal point) of the first vector, and see where we end up (new vector starts at beginning of one and ends at end of the other). If the vectors aren’t this way to begin with, we can move the second vector (as long as it has the same magnitude and direction, so it’s like a slide) to be this way. This is called the “headtotail” method.
You can think of adding vectors as connecting the diagonal of the parallelogram (a foursided figure with two pairs of parallel sides) that contains the two vectors.
Do you see how when we add vectors geometrically, to get the sum, we can just add the \(x\) components of the vector, and the \(y\) components of the vectors?
When we subtract two vectors, we just take the vector that’s being subtracting, reverse the direction and add it to the first vector. This is because the negative of a vector is that vector with the same magnitude, but has an opposite direction (thus adding a vector and its negative results in a zero vector).
Note that to make a vector negative, you can just negate each of its components (\(x\) component and \(y\) component) (see graph below).
Multiplying Vectors by a Number (Scalar)
To multiply a vector by a number, or scalar, you simply stretch (or shrink if the absolute value of that number is less than 1), or you can simply multiply the \(x\) component and \(y\) component by that number. Notice also that the magnitude is multiplied by that scalar.
Do you see how two vectors that are parallel are just a multiple of each other?
Multiplying by a negative number changes the direction of that vector.
Here’s what subtracting vectors and also multiplying vectors by a scalar looks like:
Let’s put all this together to perform the following vector operations, given the vectors shown:
You may also see problems like this, where you have to tell whether the statement is true or false. Note that you want to look at where you end up in relation to where you started to see the resulting vector. If you end up exactly where you started from, the resulting vector is 0.
Here are a couple more examples of vector problems. Notice in the second set of problems when we are given a magnitude and direction of a vector, and have to find that vector, we use the following equation, like we did when we here in the Polar Coordinates, Equations and Graphs section, where \(\left\ {\,v} \right\\) or the magnitude of a vector is like the “\(r\)” (radius) we saw for polar numbers: \(v=\left\ {\,v} \right\\left( {\cos \alpha i+\sin \alpha j} \right)=\left( {\left\ {\,v} \right\\cos \alpha } \right)i+\left( {\left\ {\,v} \right\\sin \alpha } \right)j\). (Trigonometry always seems to come back and haunt us!) We’ll We’ll leave our answers in \(ai+bj\) form.
Applications of Vectors
Vectors are extremely important in many applications of science and engineering. Since vectors include both a length and a direction, many vector applications have to do with vehicle motion and direction.
We saw above that, given a magnitude and direction, we can find the vector with \(v=\left\ {\,v} \right\\left( {\cos \alpha i+\sin \alpha j} \right)=\left( {\left\ {\,v} \right\\cos \alpha } \right)i+\left( {\left\ {\,v} \right\\sin \alpha } \right)j\), where \(\left\ {\,v} \right\\) is the speed. This way we can add and subtract vectors, and get a resulting speed and direction for the new vector.
Remember that a bearing (we saw here in the Law of Sines and Cosines, and Areas of Triangles section), is typically expressed a measure of the clockwise angle that starts due north or on the positive \(\boldsymbol{y}\)–axis (initial side) and terminates a certain number of degrees (terminal side) from that due north starting place. (This is also written, as in the case of a bearing of 40° as “40° east of north”, or “N40°E”).
(A lot of times, the bearing includes more directions, such as 70° west of north, also written as N70°W. In this case, the angle will start due north (straight up, or on the positive \(y\) axis) and go counterclockwise 70° (because it’s going west, or to the left, instead of east). Similarly, a bearing of 50° south of east, or E50°S, would be an angle that starts due east (on the positive \(x\)axis) and go clockwise 50° clockwise (towards the south, or down). Also, if you see a bearing of southwest, for example, the angle would be 45° south of west, or 225° clockwise from north, and so on.)
Each time a moving object changes course, you have to draw another line to the north to map its new bearing.
When there’s a tail wind, remember that you have to add this vector to the vector that the object is trying to go on (its programmed or “steered” course), to get the actual vector of the object. Remember:
Problem:
A plane is flying on a bearing of 25° south of west at 500 miles per hour (speed). Express the velocity of the plane (as a vector).
Solution:
Problem:
A sailboat is sailing on a bearing of 30° north of west at 6.5 miles per hour (in still water). A tail wind blowing 20 miles per hour in the direction 40° south of west alters the course of the boat.
Express the actual velocity of the sailboat as a vector. Then determine the actual speed and direction of the boat.
Solution:
Note that if we were given a vector for the actual course of the boat and had to come up with the vector for which the boat should be “steered”, we would have to subtract the wind from the actual course.
Now let’s the problem we already did using Law of Cosines from the Law of Sines and Cosines, and Areas of Triangles section:
Problem:
A cruise ship travels at a bearing of 40° at 60 mph for 3 hours, and changes course to a bearing of 120°. It then travels 40 mph for 2 hours. Find the distance the ship is from its original position and also its bearing from the original position.
Solution:
In this problem, Distance = rate x time, since we are given rates and times and need to calculate distances.
Since no specific directions (like West of South) are given for these bearings, we will obtain the angles by measuring the clockwise angle that starts due north or on the positive \(\boldsymbol{y}\)axis (East of North).
And remember that with a change of bearing, we have to draw another line to the north to map its new bearing.
Now that we have the angles, we can use vector addition to solve this problem; doing the problem with vectors is actually easier than using Law of Cosines:
Dot Product and Angle Between Two Vectors
The dot product of two vectors \(v=ai+bj\) and \(w=ci+dj\) (sort of like multiplying two vectors) is defined as \(v\bullet \,w=ac+bd\); in other words, you multiply the two “\(x\)” parts of the vectors, and multiply the two “\(y\)” parts, and then add them together. The result is a scalar (single number).
Here is an example: If \(v=2i+3j\) and \(w=2i+j\), the dot product \(v\bullet \,w=\left( {2} \right)\left( 2 \right)+\left( 3 \right)\left( 1 \right)=1\).
We use dot products to find the angle measurements between two vectors; the cosine of the angle between two vectors is the dot product of the vectors, divided by the product of each of their magnitudes:
(And we don’t need to worry about getting the correct quadrant when putting this in the calculator!)
So we might be able to use this formula instead of, say, the Law of Cosines, for applications.
Note that if the dot product of two vectors is 0, the vectors form right angles, or are orthogonal, since the cos of 90° is 0 (and thus the whole expression will be 0).
And remember that we noted above that if two vectors are parallel, then one is a “multiple” of another, or \(v=aw\). For example, the vector \(v=2i+3j\) would be parallel to the vector \(v=4i+6j\). If vectors are parallel, the angle between them is either 0 (if they are the same vector) or π.
Here are some example problems:
3D Vectors – Vectors in Space
We’ve been dealing with vectors (and everything else!) in the two dimensional plane, but “real life” is actually three dimensional, so we need to know how to work in 3D, or space, too.
A 3D coordinate system is typically drawn like this, with the zaxis going “up”. Note that the positive xaxis comes forward at you, and the positive yaxis to the right of that, if you’re looking head on. Maybe you can remember this by the expression “Exit. Why?” (x – z – y when looking head on).
Geometric Vectors in 3D are still directed line segments, but in the xyzplane. We still can find the vector between two coordinate points by “subtracting” the first vector from the second.
So, for example, if \(A\) is (–4, 2, 7) and \(B\) is (–3, 8, 0), the vector \(\overrightarrow{{AB}}\) is second point minus first point, or \(\displaystyle \left\langle {{{x}_{2}}} \right.{{x}_{1}},\,\left. {{{y}_{2}}{{y}_{1}},\,{{z}_{2}}{{z}_{1}}} \right\rangle \) or \(\left\langle {3\left( {4} \right)} \right.,\left. {82,07} \right\rangle =\left\langle {1,\left. {6,7} \right\rangle } \right.\). Note also that vectors can also be written in the form ai + bj + ck, so this vector can also be written as i + 6j –7k.
The magnitude of the 3D vector, written \(\left\ {AB} \right\\) is still the distance between the two points (like taking hypotenuse of a right triangle twice actually), or \(\sqrt{{{{{\left( {{{x}_{2}}{{x}_{1}}} \right)}}^{2}}+{{{\left( {{{y}_{2}}{{y}_{1}}} \right)}}^{2}}+{{{\left( {{{z}_{2}}{{z}_{1}}} \right)}}^{2}}}}\), or with the new vector \(\left\langle {x,} \right.\left. {y,z} \right\rangle \), it’s just \(\sqrt{{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\). So, for our points \(A\) and \(B\) above, \(\left\ {AB} \right\=\sqrt{{{{1}^{2}}+{{6}^{2}}+{{{\left( {7} \right)}}^{2}}+}}=\sqrt{{86}}\).
Vector Operations in Three Dimensions
Adding, subtracting 3D vectors, and multiplying 3D vectors by a scalar are done the same way as 2D vectors; you just have to work with three components.
Like for 2D vectors, the dot product of two vectors \(v=ai+bj+ck\) and \(w=di+ej+fk\) (sort of like multiplying two vectors) is defined as \(v\bullet \,w=ad+be+cf\); in other words, you multiply the two “\(x\)” parts of the vectors, multiply the two “\(y\)” parts, multiply the two “\(z\)” parts, and then add them together. The result is a scalar (single number).
Again, like for 2D, we use dot products to find the angle measurements between two vectors; the cosine of the angle between two vectors is the dot product of the vectors, divided by the product of each of their magnitudes:
Here are some problems; included is how to get the equation of a sphere:
Writing a 3D vector in terms of its magnitude and direction is a little more complicated. Since we can’t really describe a 3D vector in terms of only a magnitude and one direction, we have to get what we call the direction angles:
Here are what these angles look like:
It turns out for the vector \(v=ai+bj+ck\), the direction angles \(\alpha ,\,\,\beta ,\,\,\text{and}\,\,\gamma \) are:
These cosine values are called the direction cosines for the vector v.
To find the 3D vector in terms of its magnitude and direction cosines, we use:
Now let’s do a problem:
3D Vector Problem  Solution 
Find the direction angles of the following vector, and then write the vector in terms of its magnitude and direction cosines:
\(v=6i+3j2k\)

Using the equations above, we see that:
\(\displaystyle \begin{align}\cos \alpha &=\frac{a}{{\sqrt{{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}}}=\frac{a}{{\left\ {\,v} \right\}}=\frac{{6}}{7};\,\,\,\,\alpha =149{}^\circ \\\cos \beta &=\frac{b}{{\sqrt{{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}}}=\frac{3}{{\left\ {\,v} \right\}}=\frac{3}{7};\,\,\,\,\beta =64.6{}^\circ \,\,\,\,\,\\\cos \gamma &=\frac{c}{{\sqrt{{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}}}=\frac{c}{{\left\ {\,v} \right\}}=\frac{{2}}{7};\,\,\,\,\gamma =106.6{}^\circ \end{align}\) Thus, we have: \(\begin{align}v&=\left\ {\,v} \right\\left( {\left( {\cos \alpha } \right)i+\left( {\cos \beta } \right)j+\left( {\cos \gamma } \right)k} \right)\\&=\left( {7\cos 149{}^\circ } \right)i+\left( {7\cos 64.6{}^\circ } \right)j+\left( {7\cos 106.6{}^\circ } \right)k\end{align}\)

Cross Products of 3D Vectors
Also, only for vectors 3D vectors, we have what we call a cross product of vectors (also called vector product, since the result is still a vector) of two vectors \(v=ai+bj+ck\) and \(w=di+ej+fk\). The vector that is the cross product of two vectors is actually orthogonal (perpendicular) to both of the original vectors. This is also called the normal vector.
This looks crazy! But we can get this cross product using determinants of matrices. We learned about determinants of matrices here in the The Matrix and Solving Systems with Matrices section.
Remember that for a 2 by 2 matrix, we get the determinant this way:
And here is an example of how we got the determinant of a 3 by 3 matrix:
Here is an example of how we use a determinant to find the cross product of two vectors \(v=i+2j4k\) and \(w=i+5j+3k\):
The vector \(\displaystyle 26i+j+7k\) is orthogonal (perpendicular, normal) to the vectors v and w above.
A few things to remember here. First, we must watch the order of the vectors when we are finding the cross products of vectors; v × w is not necessarily the same thing as w × v.
Also, we can use the right hand rule to find the direction of the cross product of two vectors by holding up your right hand and make your index finger, middle finger, and thumb all perpendicular to each other (easier said than done!). Then point your index finger in the direction of the first vector (such as v) and your middle finger in the direction of the second vector (such as w). Your thumb will point in the direction of v × w.
This is something you probably won’t need too much in your math classes, but it can become very handy in Physics. (And remember the directions of 3D vectors as shown in the coordinate system below).
We can use the cross product to find the area of a 3D parallelogram. If that parallelogram has two adjacent sides with vectors v and w, we can take the magnitude of the vectors’ cross product to find its area: \(\left\ {v\,\,\times \,\,w} \right\\). We can also use this if given four vertices of a parallelogram; we would just have to find two adjacent sides of the parallelogram in vector form first.
Here are some cross product problems:
The Equation of a Plane
You might also be asked to find the equation of the plane that passes through a given point and is perpendicular to a certain vector, or even the equation of a plane containing three points.
Remember that the equation of a line can be in the standard form \(ax+by=c\), so the equation of a plane can be in the form \(ax+by+cz=d\). (These are called Cartesian equations.)
To see what this plane might look like, we can see where it intersects each of the three axes by setting the other variables to 0, for example with the graph \(2x+6y+3z=12\) (set y and z equal to 0 and solve for x, and so on):
Remember that the dot product will be 0 if two orthogonal vectors. It turns out that a vector equation of the plane is \(\displaystyle \left\langle {a,b,c} \right\rangle \bullet \left( {\left\langle {x,y,z} \right\rangle \left\langle {{{x}_{0}},{{y}_{0}},{{z}_{0}}} \right\rangle } \right)=0\), or \(\displaystyle \left\langle {a,b,c} \right\rangle \bullet \left\langle {x{{x}_{0}},y{{y}_{0}},z{{z}_{0}}} \right\rangle =0\), where \(\left\langle {a,b,c} \right\rangle \) is orthogonal to the plane (the normal vector) and \(\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}}} \right)\) is a point on the plane.
Another way to get the equation of the plane for a vector perpendicular to a certain plane at point \(\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}}} \right)\) is to use \(ax+by+cz=d\), where \(d=a{{x}_{0}}+b{{y}_{0}}+c{{z}_{0}}\) (plug in the point to get d).
OK, so this looks really complicated, so let’s do a problem to show it’s not too bad:
To find the equation of the plane containing three points, we first have to find two vectors defined by the points, find the cross product of the two vectors, and then use the Cartesian equation above to find d:
Parametric Form of the Equation of a Line in Space
We can get a vector form of an equation of a line in 3D space by using parametric equations. (We’ll review Parametric Equations here).
In two dimensions, we worked with a slope of the line and a point on the line (or the \(y\)intercept).
In 3D space, we can use a 3D vector \(\left\langle {a,b,c} \right\rangle \) as the slope of a line, and define that line by an initial point \(\left\langle {{{x}_{0}},{{y}_{0}},{{z}_{0}}} \right\rangle \) (vector that goes through that point and the origin) and the vector \(\left\langle {a,b,c} \right\rangle \). Note that the vector \(\left\langle {a,b,c} \right\rangle \) will be parallel to the line we’re describing, just like a slope going through the origin is parallel to a 2D line. We have three different ways to write this 3D line using parametric equations:
Notice to get the last form, we solve for \(\boldsymbol {t}\) in the second set of equations. Also note that to go from the last equation to the first equation, we have to set each to \(t\), and solve back for \(x\), \(y\), and \(z\).
This looks a little complicated, so let’s do some problems to show it’s not too bad:
Here are more problems that look more difficult, but are actually easier, since we’re finding a plane perpendicular to the vector or line:
Here are a few more problems:
Learn these rules, and practice, practice, practice!
Click on Submit (the arrow to the right of the problem) and scroll down to “Find the Angle Between the Vectors” to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.
If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software. You can even get math worksheets.
You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!
On to Parametric Equations – you are ready!