Two vectors a_vec and b_vec make an angle theta with each other. The magnitude of b_vec is half the magnitude of a_vec. Let c_vec = a_vec - b_vec and |a_vec| = a. Which of the following statements is/are correct?

if c = a*sqrt(5)/2 then theta will be 90 deg
if c = a*sqrt(3)/2 then theta will be 60 deg
if c = a/sqrt(2) then theta will be 45 deg
if c = 3a/2 then theta will be 180 deg

Correct answer: if c = a*sqrt(5)/2 then theta will be 90 deg

Solution

With |b|=a/2, we get |c|² = a² + a²/4 - a²*cos(theta) = a²*(5/4 - cos(theta)). Setting c=a*sqrt(5)/2 gives c²=5a²/4, so cos(theta)=0, theta=90 deg. The other options fail the same test.

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