Question

The projection of \(\vec{b}\) on \(\vec{a}\) is 12. If the angle between \(\vec{a}\) and \(\vec{b}\) is \(60^\circ\), then \(|\vec{b}| =\)

• A. 6
• B. 12
• C. 18
• D. 20
• E. 24

Hint:

Projection is basically the "shadow" of one vector on the other. If the shadow is 12 and the angle is \(60^\circ\), the original length must be twice the shadow length since \(\cos 60^\circ = 0.5\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The scalar projection of one vector onto another is the product of the first vector's magnitude and the cosine of the angle between them.

Step 2: Key Formula or Approach:
Projection of \(\vec{b}\) on \(\vec{a} = |\vec{b}| \cos \theta\).

Step 3: Detailed Explanation:
Given:
Projection = 12
\(\theta = 60^\circ\)
Using the formula:
\[ 12 = |\vec{b}| \cos 60^\circ \]
\[ 12 = |\vec{b}| \left( \frac{1}{2} \right) \]
\[ |\vec{b}| = 12 \times 2 = 24 \]

Step 4: Final Answer:
The magnitude of \(\vec{b}\) is 24.