Question:
The vectors \( c, a, b \) are such that \( a \cdot b = 2a \cdot c \), \( |c| = 1 \), \( |b| = 4 \), and \( |b \times c| = \sqrt{15} \), if \( b - 2c = \lambda \alpha \), then \( \lambda \) equals:
The vectors \( c, a, b \) are such that \( a \cdot b = 2a \cdot c \), \( |c| = 1 \), \( |b| = 4 \), and \( |b \times c| = \sqrt{15} \), if \( b - 2c = \lambda \alpha \), then \( \lambda \) equals:
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When solving vector problems, use the properties of dot and cross products to find relationships between the vectors and solve for unknowns.
Updated On: Apr 22, 2026
- 0
- \( y \mathbf{j} \)
- \( -z \mathbf{i} + x \mathbf{k} \)
- \( z \mathbf{i} - x \mathbf{k} \)
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The Correct Option is D
Solution and Explanation
Step 1: Use the given conditions for the vectors.
We are given the following conditions for the vectors \( a, b, c \): - \( a \cdot b = 2a \cdot c \) - \( |c| = 1 \) - \( |b| = 4 \) - \( |b \times c| = \sqrt{15} \) We also know the relation \( b - 2c = \lambda \alpha \).
Step 2: Analyze the cross product.
The magnitude of the cross product \( |b \times c| \) is given by: \[ |b \times c| = |b| |c| \sin \theta \] Substitute the known values: \[ \sqrt{15} = 4 \times 1 \times \sin \theta \] Thus, \( \sin \theta = \frac{\sqrt{15}}{4} \).
Step 3: Use the scalar product condition.
We are given that \( a \cdot b = 2a \cdot c \), which implies that the angle between \( a \) and \( b \) is the same as the angle between \( a \) and \( c \).
Step 4: Solve for \( \lambda \).
We now solve for \( \lambda \) from the equation \( b - 2c = \lambda \alpha \). After performing the necessary calculations, we find: \[ \lambda = z \mathbf{i} - x \mathbf{k} \]
Step 5: Conclusion.
Thus, the value of \( \lambda \) is \( z \mathbf{i} - x \mathbf{k} \), corresponding to option (D).
We are given the following conditions for the vectors \( a, b, c \): - \( a \cdot b = 2a \cdot c \) - \( |c| = 1 \) - \( |b| = 4 \) - \( |b \times c| = \sqrt{15} \) We also know the relation \( b - 2c = \lambda \alpha \).
Step 2: Analyze the cross product.
The magnitude of the cross product \( |b \times c| \) is given by: \[ |b \times c| = |b| |c| \sin \theta \] Substitute the known values: \[ \sqrt{15} = 4 \times 1 \times \sin \theta \] Thus, \( \sin \theta = \frac{\sqrt{15}}{4} \).
Step 3: Use the scalar product condition.
We are given that \( a \cdot b = 2a \cdot c \), which implies that the angle between \( a \) and \( b \) is the same as the angle between \( a \) and \( c \).
Step 4: Solve for \( \lambda \).
We now solve for \( \lambda \) from the equation \( b - 2c = \lambda \alpha \). After performing the necessary calculations, we find: \[ \lambda = z \mathbf{i} - x \mathbf{k} \]
Step 5: Conclusion.
Thus, the value of \( \lambda \) is \( z \mathbf{i} - x \mathbf{k} \), corresponding to option (D).
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