Question

Let \( a, b, c \) be the three vectors such that \( c \neq 0 \) and \[ a \cdot b = 2a \cdot c, \quad |c| = |c| = 1, \quad |b| = 4, \quad |b \times c| = \sqrt{15}, \quad \text{if} \quad b - 2c = \lambda \alpha, \quad \text{then} \quad \lambda \quad \text{equals} \]

• A. \( -1 \)
• B. \( 2 \)
• C. \( -4 \)
• D. \( 3 \)

Hint:

When dealing with vector equations, use the properties of dot and cross products to relate the angles between vectors and solve for unknowns.

Answer 1

The Correct Option is C

Step 1: Use the given information.
We are given the conditions: \[ a \cdot b = 2a \cdot c, \quad |c| = 1, \quad |b| = 4, \quad |b \times c| = \sqrt{15} \] We also know the relation \( b - 2c = \lambda \alpha \).

Step 2: Use vector cross product properties.
We can use the property of the cross product: \[ |b \times c| = |b| |c| \sin \theta \] Given \( |b \times c| = \sqrt{15} \) and \( |b| = 4 \), \( |c| = 1 \), we substitute: \[ \sqrt{15} = 4 \times 1 \times \sin \theta \] Thus, \( \sin \theta = \frac{\sqrt{15}}{4} \).

Step 3: Use the scalar product information.
We are also given: \[ a \cdot b = 2a \cdot c \] This implies that the angle between \( a \) and \( b \) is the same as the angle between \( a \) and \( c \).

Step 4: Calculate the value of \( \lambda \).
Using the relation \( b - 2c = \lambda \alpha \), we calculate \( \lambda \) by solving the cross product relation. The result gives us: \[ \lambda = -4 \]

Step 5: Conclusion.
Thus, the value of \( \lambda \) is \( -4 \), which corresponds to option (C).