Note that vectors can be 1, 2, or 3-dimensional. Thus, vectors can have an x component, an x and y component, or an x, y, and z component. Let’s say that we have two 3-dimensional vectors, vector A and vector B. We might write these vectors in components as A = <Ax,Ay,Az > and B = <Bx,By,Bz>, using x y z components accordingly.

In general terms, A+B = <Ax+Bx,Ay+By,Az+Bz>. Let’s add two vectors A and B. Example: A = <5, 9, -10> and B = <17, -3, -2>. A + B = <5+17, 9+-3, -10+-2>, or <22, 6, -12>.

In general terms, A-B = <Ax-Bx,Ay-By,Az-Bz> Let’s subtract two vectors A and B. A = <18, 5, 3> and B = <10, 9, -10>. A - B = <18-10, 5-9, 3-(-10)>, or <8, -4, 13>.

When making a scale drawing of a vector, you must take care to measure and draw all angles accurately. Mis-drawn angles will lead to poor answers.

Note that the order you join the vectors in is not important. Vector A + Vector B = Vector B + Vector A

If you drew all of your vectors to scale, measuring all angles exactly, you can find the magnitude of the resultant vector by measuring its length. You can also measure the angle that the resultant makes with either a specified vector or the horizontal/vertical etc. to find its direction. If you didn’t draw all vectors to scale, you probably need to calculate the magnitude of the resultant using trigonometry. You may find the Sine Rule and the Cosine Rule helpful here. [9] X Research source If you are adding more than two vectors together, it is helpful to first add two, then add their resultant with the third vector, and so on. See the following section for more information.

For example, if the vectors we added represented velocities in ms-1, we might define our resultant vector as “a velocity of x ms-1 at yo to the horizontal”.

The lengths of the two sides are equal to the magnitudes of the x and y components of your vector and may be calculated using trigonometry. [13] X Research source If x is the magnitude of the vector, the side adjacent to the vector’s angle (relative to the horizontal, vertical, etc. ) angle is xcos(θ), while the side opposite is xsin(θ). It’s also important to note the direction of your components. If the component points in the negative direction of one of your axes, it is given a negative sign. For example, in a 2-D plane, if a component points to the left or downwards, it is given a negative sign. For example, let’s say that we have a vector with a magnitude of 3 and a direction of 135o relative to the horizontal. With this information, we can determine that its x component is 3cos(135) = -2. 12 and its y component is 3sin(135) = 2. 12

For instance, let’s say that our vector from the previous step, <-2. 12, 2. 12>, is being added to the vector <5. 78, -9>. In this case, our resultant vector would be <-2. 12+5. 78, 2. 12-9>, or <3. 66, -6. 88>.

To find the magnitude of the vector whose components we found in the previous step, <3. 66, -6. 88>, let’s use the Pythagorean Theorem. Solve as follows: c2=(3. 66)2+(-6. 88)2 c2=13. 40+47. 33 c=√60. 73 = 7. 79

To find the direction of our example vector, let’s use θ=tan-1(b/a). θ=tan-1(-6. 88/3. 66) θ=tan-1(-1. 88) θ=-61. 99o

For example, if our example vector represented a force (in Newtons), then we might write it as “a force of 7. 79 N at -61. 99o to the horizontal”.