Vector Algebra

6 Vector Algebra

We have learned to add vectors. But can we subtract vectors?

Consider any old vector $\vec r$ . Whatever subtraction means, it must be true that $\vec r - \vec r = \vec 0$ . Here, the zero vector means “I didn’t go anywhere.” Just as with numbers, we can think of subtraction as “adding a negative vector,” which results in the expression $\vec r + (-\vec r) = \vec 0$ . Evidently, $-\vec r$ is identical to $\vec r$ , but points in the opposite direction. This is a key lesson we will use over and over:

A “minus” sign means “in the opposite direction.”

Example

Consider the vectors $\vec r$ and $\vec s$ below. Find the vector $\vec s - \vec r$ .

Vector subtraction can be useful for solving vector problems. Let’ do an example.

Exercise 6.1: Using Vector Subtraction

Serena walks 5 mi in the NE direction. She then walks west. After a short while, Serena heads back home. Her last leg of the journey takes 4 mi. How much distance did Serena cover during the second leg of her journey?

If you get stuck, you can reveal my diagram by moving the slider. Then find relations. At this point you can use vector subtraction to isolate the unknown vector.

If you’re still stuck, see my “find relations” solution.

Now we can use the fact that a vector equation is really one equation per component to finish solving the problem.

As it turns out, we can also multiply a vector by a number. In math, this is referred to as scalar multiplication, because it “scales” a vector. For instance, $2\vec A = (2A_x,2A_y)$ points in the same direction as $\vec A$ , but is twice as long. The components of $2\vec A$ are just the components of $\vec A$ multiplied by $2$ , so that makes sense. Finally, note $\vec A +\vec A = 2\vec A$ : everything basically works like you’d expect!

More generally, the components of $\lambda \vec A$ are just the components of $\vec A$ multiplied by $\lambda$ . This is,

$\lambda \vec A = (\lambda A_x,\lambda A_y)$ .

This is probably what you would have guessed happens, which is one of the great virtues of vector algebra: vector addition and scalar multiplication behaves pretty much as you’d expect.

Because regular old numbers can “scale” vectors up and down, physicists tend to refer to plain old numbers as scalars. Why people felt the need for inventing another name for “numbers” is completely beyond me, but people use the term “scalar” all the time, so you have to know it.

Let’s try two quick problems.

Exercise 6.2: Vector Algebra

You are given three vectors: $\vec A=(2,3)$ , $\vec B = (1,-4)$ , and $\vec C = (-3,3)$ .

Find the vectors: $\vec r_1 = 2\vec A +3\vec B$ ; $\vec r_2 = 2\vec B - \vec C$ ; and $\vec r_3 = -3\vec A + 2\vec C$ .

If you get stuck, I include a detailed solution for how to find $\vec r_3$ below. Make sure you are able to solve the remaining ones!

Exercise 6.3: Scalar Multiplication

Show that given any vector $\vec A=(A_x,A_y)$ , the magnitude of the vector $\lambda |\vec A|$ is just $|\lambda||\vec A|$ . That is, we can “pull out” the $\lambda$ from the magnitude sign, provided we also take the absolute value of $\lambda$ .

Use your results to rank-order the vectors below by increasing magnitude, smallest at the left, and largest on the right. Not every column needs to contain a vector, and each column may contain more than one vector. The vector $\vec A$ is an arbitrary non-zero vector.

Key Takeaways

6 Vector Algebra

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