Question

A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \lambda \hat{k}$, then the value of $\lambda$ is:

• A. 6
• B. 1
• C. $\frac{1}{4}$
• D. 4

Hint:

When vectors are parallel, their components must be proportional. Use this fact to solve for unknowns.

Answer 1

The Correct Option is B

The parallel nature of the ropes implies that the two vectors are scalar multiples. We formulate the equation: \[ 3\hat{i} + 15\hat{j} + 6\hat{k} = k(2\hat{i} + 10\hat{j} + \lambda \hat{k}) \] Component-wise equality yields: \[ 3 = 2k, \quad 15 = 10k, \quad 6 = \lambda k \] The second equation gives $k = 1.5$. Substituting this value into the third equation results in: \[ 6 = \lambda \times 1.5 \quad \Rightarrow \quad \lambda = 4 \] Consequently, $\lambda = 4$.