Math question

Question

V=3i+6j

find ||-5v||

 

A. 15 square root 3

B. 15 square root 5 

C. -15 square 5

D. 15i square root 3

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Answer 1

There are a few ways to approach this question:

 

1. Straightforward way:

 

If v = 3i + 6j, then -5v = -5*(3i + 6j) = -15i-30j.

Then, ||-5v|| = sqrt[(-15)^2 + (-30)^2] = sqrt(225 + 900) = sqrt(1125) = sqrt[225*5] = sqrt(225)*sqrt(5) = 15*sqrt(5), so the answer is B.

 

2. Using properties of magnitudes of vectors:

 

When you multiply a vector by a constant, the magnitude of the vector is also multiplied by the absolute value of that constant. In this case, the magnitude is multiplied by |-5| = 5. So all we have to do is find |v| and multiply it by 5.

|v| = |3i + 6j| = sqrt(3^2 + 6^2) = sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5).

Multiplying this by 5, we have ||-5v|| = 15*sqrt(5) as before.

 

Note, that by definition, the magnitude of a vector is always a non-negative real number, so choices C and D can be eliminated immediately without doing any calculations since choice C is a negative real number and choice D is an imaginary number.