Question

The angle between vectors \(A = 2i - j + 2k\) and \(B = 6i - 3j + 6k\) is:

• A. 0
• B. 30
• C. 45
• D. 60

Hint:

Check whether one vector is simply a scalar multiple of the other before computing the dot product.

Answer 1

The Correct Option is A

Step 1: Notice a pattern first: vector \(B = 6i - 3j + 6k\) can be written as \(3(2i - j + 2k) = 3A\). So B is simply a positive scalar multiple of A.
Step 2: Geometrically, when one vector is a positive scalar multiple of another, both vectors point in exactly the same direction, so the angle between them must be \(0^\circ\).
Step 3: Confirm using the dot product formula \(\cos\theta = \frac{A\cdot B}{|A||B|}\). Compute the dot product: \(A\cdot B = (2)(6) + (-1)(-3) + (2)(6) = 12 + 3 + 12 = 27\).
Step 4: Compute the magnitudes: \(|A| = \sqrt{4+1+4} = 3\), and \(|B| = \sqrt{36+9+36} = 9\).
Step 5: So \(\cos\theta = \frac{27}{27} = 1\), which gives \(\theta = \cos^{-1}(1) = 0^\circ\). None of 30, 45 or 60 degrees give a cosine of exactly 1, so the answer is 0, option 1.