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Points, Vectors, and Functions

Points, Vectors, and Functions

At a Glance - Scaling Vectors

Quiz time. Already? Yep.

What's one apple plus another apple? Two apples.

What's two oranges less one orange? One orange.

What's one griffin plus one minotaur? Any mythology buff could tell you it's a griffotaur, of course.

Adding and subtracting vectors is something like adding the griffin and the minotaur. It's possible to add two vectors together or subtract one vector from another. We get another vector in its place. To add vectors, we add the x-components to each other and add the y-components to each other.

Sample Problem

To add the vectors <3, 4> and <5, 6> we add the first components together and add the second components together:

< 3, 4> + <5, 6 > = <8, 10>.

Sample Problem

To subtract, we subtract the x-components and then subtract the y-components.

<3, 4> – <5, 6> = <-2, -2>.

When we add or subtract vectors, remember to deal with the x-components and the y-components separately. Why? Going back to counting fruit, apples are the x-component and oranges are the y-component of vectors. We can't add apples to oranges, but we can add apples to apples and oranges to oranges.

Now we're going to work with multiplication of a vector by a real number. To multiply a vector by a number, we multiply each component of the vector by that number.

Multiply the minotaur vector by 2. What do we get? A minotaur twice the size. Likewise, if we multiply the griffin by 0.01, we get a pint-sized, cute griffin to take home with us.

When we multiply a vector by a real number, we call the number a scalar.

We call the real number a scalar because it scales the vector to be a different size. If the scalar is negative it will also change the direction of the vector.

Sample Problem

Take the vector <1, 2>. This vector has magnitude √5

Now multiply the vector by 3 to get a new vector: <1, 2> → <3, 6>. The magnitude of the new vector is

The original vector <1, 2> had magnitude √5, The magnitude of the new vector <3, 6> is 3 times greater than the magnitude of the original vector, <1, 2>. In multiplying the vector by 3, we scaled the arrow to be 3 times longer.

Example 1

Multiply the vector <9, 2> by 5.

Example 2

What is the magnitude of the vector 5<2, 3>?

Example 3

If the vector a<4, 3> has magnitude 10, what is a?

Example 4

By what number must we multiply the vector <3, 5> to get a vector of length 2?

Example 5

What happens to the magnitude and direction of the vector <2, 2> when it is multiplied by -1 ?

Exercise 1

Perform the following vector operation. 

  • <2, -10> + <4, 6>

Exercise 2

What is the sum of these two vectors?

  • <3, 7> + <-2, -3>

Exercise 3

What's the difference between these two vectors?

  • <1, 1> – <-3, -6>

Exercise 4

Find the sum of the following two vectors.

  • <2, 2> + <1, 5>

Exercise 5

Perform the vector operation.

  • <11, 0.3> – <1, 0.4>

Exercise 6

What is the resulting vector?

  • 7<3, 4>

Exercise 7

What is the resulting vector after performing the multiplication?

  • 2<2, 8>

Exercise 8

Perform the scalar multiplication seen here.

  • -5<-3, -7>

Exercise 9

Work out the multiplication problem. 

43<6, -1>

Exercise 10

What is the resulting vector after performing the scalar multiplication?

  • -3 <6, -1>

Exercise 11

What vector are we left with after doing the following scalar multiplication?

  • 0<7, 6>

Exercise 12

  • Find ||3<7, 10>||.

Exercise 13

  • A vector v has magnitude 10. What is the magnitude of the vector  ?

Exercise 14

  • If the vector a<5, 12> has magnitude 39, what is a?

Exercise 15

  • Find the magnitude of the vector (-1)<8, 3>.

Exercise 16

  • Find the magnitude of the vector -2<3, 2>.

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