Math: Vectors and Vector Operations Worksheet

Question 1

Vector a = <3, 4>. Find the magnitude of vector a.

Accepted Answer

The magnitude of vector a is 5 because sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

Question 2

Let u = <2, -1> and v = <5, 3>. Find u + v.

Accepted Answer

The sum is <7, 2> because you add corresponding components: 2 + 5 = 7 and -1 + 3 = 2.

Question 3

Let u = <6, 2> and v = <1, 7>. Find u - v.

Accepted Answer

The difference is <5, -5> because you subtract corresponding components: 6 - 1 = 5 and 2 - 7 = -5.

Question 4

Find 3< -2, 5 >.

Accepted Answer

The result is <-6, 15> because each component is multiplied by 3.

Question 5

A vector has initial point (1, 2) and terminal point (6, 8). Write the vector in component form.

Accepted Answer

The vector is <5, 6> because 6 - 1 = 5 and 8 - 2 = 6.

Question 6

A plane travels 120 km east and then 50 km north. Represent the trip as a vector in component form.

Accepted Answer

The vector is <120, 50> if east is positive x and north is positive y.

Question 7

Find the magnitude of v = <8, 15>.

Accepted Answer

The magnitude is 17 because sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17.

Question 8

Let p = <-4, 9> and q = <3, -2>. Find 2p + q.

Accepted Answer

First, 2p = <-8, 18>. Then 2p + q = <-8, 18> + <3, -2> = <-5, 16>.

Question 9

Determine whether the vectors <4, 6> and <2, 3> are scalar multiples of each other. Explain briefly.

Accepted Answer

Yes, they are scalar multiples because <4, 6> = 2<2, 3>. Both components are multiplied by the same number, 2.

Question 10

Find the midpoint of the segment with endpoints A(2, -3) and B(10, 5).

Accepted Answer

The midpoint is (6, 1) because ((2 + 10) / 2, (-3 + 5) / 2) = (6, 1).

Question 11

A hiker walks 7 miles west and 24 miles south. Write the displacement vector in component form and find its magnitude.

Accepted Answer

The displacement vector is <-7, -24> if east is positive x and north is positive y. Its magnitude is 25 miles because sqrt((-7)^2 + (-24)^2) = sqrt(49 + 576) = sqrt(625) = 25.

Question 12

Let a = <1, 2> and b = <-3, 4>. Find a + 2b.

Accepted Answer

First, 2b = <-6, 8>. Then a + 2b = <1, 2> + <-6, 8> = <-5, 10>.

Question 13

A vector has magnitude 10 and points directly along the positive x-axis. Write the vector in component form.

Accepted Answer

The vector is <10, 0> because all of its length is in the positive x-direction and none is in the y-direction.

Question 14

Find the distance between the points ( -1, 4 ) and ( 5, -4 ).

Accepted Answer

The distance is 10 because the change in x is 6 and the change in y is -8, so the distance is sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10.

Question 15

Vector u = and vector v = <3, 2>. If u + v = <11, 9>, find x.

Accepted Answer

The value of x is 8 because + <3, 2> = . Since x + 3 = 11, x = 8.

Math: Vectors and Vector Operations

Math: Vectors and Vector Operations

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