Source: Extra Practice

The angle between the two vectors A⃗=3i^+4j^+5k^\vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} and B⃗=3i^+4j^−5k^\vec{B} = 3\hat{i} + 4\hat{j} - 5\hat{k} is:

Options

Option A is correct

90∘90^\circ

Option B

0∘0^\circ

Option C

180∘180^\circ

Option D

cos⁡−1(0.5)\cos^{-1}(0.5)

Explanation

To find the angle between the two vectors, we can use the dot product formula: A⃗⋅B⃗=ABcos⁡θ\vec{A} \cdot \vec{B} = AB \cos\theta. Let's compute the dot product: A⃗⋅B⃗=(3)(3)+(4)(4)+(5)(−5)=9+16−25=0\vec{A} \cdot \vec{B} = (3)(3) + (4)(4) + (5)(-5) = 9 + 16 - 25 = 0. Since the dot product is zero, the angle θ\theta between them must satisfy cos⁡θ=0\cos\theta = 0, which means θ=90∘\theta = 90^\circ. The other options are incorrect because a non-zero dot product is required for any angle other than 90∘90^\circ.