Question

The vector that must be added to \[ \mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \quad \text{and} \quad 3\mathbf{i} + 6\mathbf{j} - 7\mathbf{k} \] so resultant vector is a unit vector along the X-axis is:

• A. \( 4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} \)
• B. \( -4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} \)
• C. \( 3\mathbf{i} + 5\mathbf{k} \)
• D. \text{Null vector}

Hint:

For unit vectors, ensure that the resulting vector has only one non-zero component and its magnitude is 1.

Answer 1

The Correct Option is B

We need the resultant vector to be a unit vector along the X-axis. This means the Y and Z components of the resultant vector must be zero. Adding the two vectors: \[ \mathbf{A} = (1 - 3)\mathbf{i} + (-3 + 6)\mathbf{j} + (2 - 7)\mathbf{k} = -2\mathbf{i} + 3\mathbf{j} - 5\mathbf{k} \] Now, add a vector \( \mathbf{B} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) so that the resultant vector has no Y or Z components and its magnitude is 1. For the X-component, \( -2 + x = 1 \Rightarrow x = 3 \). For the Y-component, \( 3 + y = 0 \Rightarrow y = -3 \). For the Z-component, \( -5 + z = 0 \Rightarrow z = 5 \). Thus, the vector to be added is \( \mathbf{B} = 3\mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \). Thus, the correct answer is \( -4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} \).