Chapter 2: Vectors

2.3 Vector arithmetic and algebra

2.3.1 Scalar multiplication

You can multiply a scalar by a vector by simply using the distributive property:

\[ \begin{align}
c\vec{A} &= c(A_x\hat{x} + A_y\hat{y}) \\
&= (cA_x)\hat{x} + (cA_y)\hat{y} \tag{2.2}
\end{align} \]

for some scalar \(c\). This is called scaling the vector—hence the name scalar.

Example

You may use the interactive below to observe how scalar multiplication works. You can adjust the x and components of the vector, and also adjust the scalar c. After playing around with the interactive to get a sense of how it works, you should make some predictions: write down a vector on a piece of paper, and multiply it by some scalar. Then check your answer using the interactive.

Practice

2.3.2 Vector addition and subtraction

You will frequently need to add vectors together. There are two methods of adding vectors: graphically and algebraically.

Graphically

Place the “tail” of one vector at the origin. Place the “tail” of the other at the “tip” of the first. The sum of the two is the vector that points from the “tail” of the first vector to the “tip” of the second.

For subtraction, keep the magnitude (length) the same, but put it in the opposite direction.

This is called tip-to-tail vector addition. It is very easy to visualize the vector sum this way, but it is cumbersome to use in practice.

Algebraically

The figure below shows an example of tip-to-tail addition, with the components of each vector explicitly shown. You should notice that the \(x\) component of the sum of two vectors is the sum of each vector’s \(x\) component, and the \(y\) component of the sum of the two vectors is the sum of each vector’s \(y\) component.

Two vectors are added tip-to-tail on a Cartesian coordinate grid. The x and y components are shown using dashed lines for each vector, and for the sum of the two vectors.
The top-to-tail vector addition with vector components shown explicitly.

When adding vectors, simply add each component; this is the associative property of addition. If we have vectors \(\vec{A}\) and \(\vec{B}\) given in component form as

\[\begin{align}
\vec{A} &= A_x\hat{x} + A_y\hat{y} \\
\vec{B} &= B_x\hat{x} + B_y\hat{y}
\end{align}
\]

Then their sum (call it vector \(\vec{C}\)) is:

\[
\begin{align}
\vec{C} &= \vec{A} + \vec{B} \\
&= \left( A_x\hat{x} + A_y\hat{y} \right) + \left( B_x\hat{x} + B_y\hat{y}\right) \\
&= A_x\hat{x} + B_x\hat{x} + A_y\hat{y} + B_y\hat{y} \\
&= \underbrace{\left( A_x + B_x \right)}_{C_x}\hat{x} + \underbrace{\left( A_y + B_y \right)}_{C_y}\hat{y} \\
&= C_x\hat{x} + C_y\hat{y} \\
\end{align}
\]

Practice

Example

Say vector \(\vec{A}\) has a magnitude of 5 cm, and vector \(\vec{B}\) has a magnitude of 3 cm. Can you find the magnitude of the sum of these two vectors?

No, you would also need to know the direction of each vector. You can prove it to yourself: take some graph paper, a ruler, and an protractor. Make two vectors with the magnitudes that are given in this problem, and use any random angle for the direction. Then add the two together. You’ll find two things:

  1. The magnitude of the sum of the two vectors is not the same as the sum of the magnitude of each vector
  2. If you do this again with different angles, you will find a different resultant vector.