Vector Operations

If c=<-5, 4> and d=<8, 0>, calculate 2c+d

• <-2, 8>

• <-2, 0>

• <6, 8>

• <6, 0>

Answer explanation

To calculate 2c+d, we multiply each component of c by 2, then add the corresponding components of d. This gives us <2*(-5)+8, 2*4+0> = <-10+8, 8> = <-2, 8>, which matches the correct answer choice.

If w = <2, -5> and y = <2, 0>, find 2w + y.

• <-10, 2>

• <4, -5>

• <-6, 10>

• <6, -10>

Answer explanation

To find 2w + y, multiply each component by 2 and then add the corresponding components. 2w = <4, -10> and 2w + y = <6, -10>, which is the correct answer.

If y = <2, 0> and
z = <-1, -4>, find 3y - 2z.

• <5, -8>

• <4, 8>

• <8, 8>

• <-8, -8>

Answer explanation

To find 3y - 2z, multiply each component of y by 3 and each component of z by 2, then subtract the results. This gives <8, 8>, which is the correct answer.

u + v

• 4i - j

• 6i + j

• 3j + 4j

• i + 6j

Answer explanation

The correct answer is 6i + j because u + v is equal to 2i + 4i + j = 6i + j.

Given u = <3, 7> and v =<-5, 4>, find 2u - 3v

• <21, 2>

• <-11, 26>

• <8, 3>

• 23

Answer explanation

To find 2u - 3v, multiply each component of u and v by their respective scalars, then subtract. 2u = <6, 14> and 3v = <-15, 12>. Subtracting gives <21, 2>, which is the correct answer.

If a turtle moves from coordinate
 P = (3,2) to
Q = (5,6).  Which vector models the turtle's motion?

• 〈-2,-4〉

• 〈2,4〉

• 〈8,8〉

• none of these

Answer explanation

The vector 〈2,4〉 represents the turtle's motion from P(3,2) to Q(5,6) by moving 2 units in the x-direction and 4 units in the y-direction.

Given v = 〈3,-5〉 and w = 〈-2,3〉, what is v + w?

• 〈6,-10〉

• 〈1,-2〉

• 〈0,-1〉

• none of these

Answer explanation

To find v + w, add the corresponding components of v and w: (3 + (-2), -5 + 3) = (1, -2). Therefore, v + w = <1, -2>.

Find the components of a vector v such that 
v + 〈-6,4〉 = 〈10,-3〉

• 〈4,1〉

• 〈16,4〉

• 〈16,-7〉

• none of these

Answer explanation

To find the components of vector v, subtract -6 from 10 to get 16 for the x-component, and subtract 4 from -3 to get -7 for the y-component. Therefore, the components of vector v are <16, -7>.

Given 
v = 〈3,-5〉 and 
w = 〈-2,3〉, what is 2v + 3w?

• 〈6,-10〉

• 〈1,-2〉

• 〈0,-1〉

• none of these

Answer explanation

To find 2v + 3w, multiply each vector by its respective scalar and add the results. 2v = 2(3,-5) = (6,-10) and 3w = 3(-2,3) = (-6,9). Adding these gives (6,-10) + (-6,9) = (0,-1). Therefore, the correct answer is 〈0,-1〉.

Find the magnitude of the vector -5i + 12j

• 13

• 10.91

• 7

• 17

Answer explanation

The magnitude of the vector -5i + 12j is calculated using the Pythagorean theorem: sqrt((-5)^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. Therefore, the correct magnitude is 13.