Adding and subtracting vectors

Head to tail

Narrated video. Transcript:

An auto driver goes 4 km east, then 3 km north. How far is she from where she started, and in which direction?

Draw the first trip as arrow A, 4 units long, pointing east. Now place the tail of arrow B, the second trip, at the head of A.

Draw an arrow from the tail of A to the head of B. That arrow is the sum, R = A + B. This is the head-to-tail, or triangle, method.

Measure it: R is 5 km long, about 37° north of east. She drove 7 km in all, but her displacement is only 5 km.

Now swap the order: 3 km north first, then 4 km east. Watch the second triangle. It ends at exactly the same point. So A + B = B + A. Vector addition is commutative.

Put the two triangles together and you get a parallelogram. That gives a second method: draw A and B from the same point, complete the parallelogram, and the diagonal from that point is R.

With three vectors, you can add A and B first, or B and C first. The final arrow is the same: (A + B) + C = A + (B + C). Vector addition is associative.

The null vector and subtraction

Narrated video. Transcript:

Walk 20 m to the canteen and 20 m straight back. Draw it as arrow A, then arrow −A starting from the head of A.

The head of the second arrow lands right back on the tail of the first. The sum has zero length. We call it the null vector, or zero vector, written 0.

Adding the null vector changes nothing: A + 0 = A. Multiplying any vector by zero gives the null vector, and so does multiplying the null vector by any number.

You get the null vector whenever something returns to where it started. After one full lap of a running track, the runner’s displacement is zero.

Now subtraction. To find A − B, turn it into an addition: A − B = A + (−B).

Flip B to get −B, the same length but pointing the other way. Place −B head to tail after A. The arrow from the tail of A to the head of −B is A − B.

Here is a shortcut. Draw A and B from the same point. The arrow from the tip of B to the tip of A is also A − B.

Notice that A − B and B − A are the same length but point opposite ways. Subtraction is not commutative.

Adding with a formula

Narrated video. Transcript:

Drawings are great, but measuring with a ruler and protractor is slow and rough. Let’s find the resultant with trigonometry instead.

Draw A and B from point O, with angle θ between them, and complete the parallelogram. The diagonal is R.

From the far corner, drop a perpendicular onto the line of A, extended. Watch B split into two parts: B cos θ along A, and B sin θ across it.

Now we have a right-angled triangle. Its base is A + B cos θ and its height is B sin θ. Pythagoras gives R² = (A + B cos θ)² + (B sin θ)².

Expand it and use sin²θ + cos²θ = 1. You get R = √(A² + B² + 2AB cos θ). This is the law of cosines.

The same triangle gives the direction. If α is the angle between R and A, then tan α = B sin θ / (A + B cos θ).

Check the extremes. At θ = 0 the arrows line up and R = A + B, the largest it can be. At θ = 180° they oppose and R = |A − B|, the smallest. At 90°, R = √(A² + B²).

For our auto: A = 4 km, B = 3 km, θ = 90°. R = √(16 + 9) = 5 km, and tan α = 3/4, so α is about 37°. The same answer as the drawing.

Try it: add two vectors

Interactive simulation. Things to try:

• Drag the blue tip to change A and the green tip to change B. Amber is R.
• Turn on the parallelogram method and compare it with head to tail.
• Check that R always matches √(A² + B² + 2AB cos θ).
• Switch to Subtract and see −B flip around.

Quick check

Two forces of 3 N and 4 N act on a box. Which of these CANNOT be the magnitude of their resultant?

1. 1 N
2. 5 N
3. 7 N
4. 8 N

Using the formula

Two forces, each of 5 N, act on a point with an angle of 120° between them. What is the magnitude of the resultant?

1. 0 N
2. 5 N
3. 8.66 N
4. 10 N

Summary

• Magnitude of R = A + B: R = √(A² + B² + 2AB cos θ)
• Direction of R (angle α with A): tan α = B sin θ / (A + B cos θ)
• Subtraction: A − B = A + (−B)
• Laws of addition: A + B = B + A, (A + B) + C = A + (B + C)

• To add, place the vectors head to tail. The sum runs from the first tail to the last head. The parallelogram method gives the same answer.
• The null vector 0 has zero magnitude: A + (−A) = 0 and A + 0 = A.
• To subtract B, add −B. A − B and B − A are equal in size but opposite in direction.
• R lies between |A − B| (θ = 180°) and A + B (θ = 0). At θ = 90°, R = √(A² + B²).