Question

If \( \lambda(3\hat{i} + 2\hat{j} - 6\hat{k}) \) is a unit vector, then the values of \( \lambda \) are:

• A. \( \pm \frac{1}{7} \)
• B. \( \pm 7 \)
• C. \( \pm \sqrt{43} \)
• D. \( \pm \frac{1}{\sqrt{43}} \)
• E. \( \pm \frac{1}{\sqrt{7}} \)

Hint:

Always remember that \( \lambda \) can be positive or negative because the magnitude involves an absolute value. Squaring the components of the vector first helps avoid sign errors.

Answer 1

The Correct Option is A

Concept: A unit vector is defined as a vector whose magnitude (length) is exactly 1. For a vector \( \vec{A} \), the magnitude is \( |\vec{A}| = 1 \). If a scalar \( \lambda \) is multiplied by a vector, the magnitude becomes \( |\lambda| \times |\text{vector}| \).

Step 1: Calculate the magnitude of the given vector part.
Let the vector be \( \vec{v} = 3\hat{i} + 2\hat{j} - 6\hat{k} \). \[ |\vec{v}| = \sqrt{(3)^2 + (2)^2 + (-6)^2} \] \[ |\vec{v}| = \sqrt{9 + 4 + 36} \] \[ |\vec{v}| = \sqrt{49} = 7 \]

Step 2: Set the total magnitude to 1.
The given vector is \( \lambda\vec{v} \). For this to be a unit vector: \[ |\lambda\vec{v}| = 1 \] \[ |\lambda| \times 7 = 1 \]

Step 3: Solve for \( \lambda \).
\[ |\lambda| = \frac{1}{7} \] \[ \lambda = \pm \frac{1}{7} \]