MHTCET 10 May 2024 Shift 2 Paper

Concept:A vector parallel to a given vector is a scalar multiple of that vector.The scalar is chosen so that the resulting vector has the required magnitude.Explanation:First, compute the vector $2\bar{a} - \bar{b} + 3\bar{c}$.Substitute $\bar{a} = \hat{i} + \hat{j} + \hat{k}$, $\bar{b} = 4\hat{i} - 2\hat{j} + 3\hat{k}$, and $\bar{c} = \hat{i} - 2\hat{j} + \hat{k}$:$$2\bar{a} - \bar{b} + 3\bar{c} = (2-4+3)\hat{i} + (2+2-6)\hat{j} + (2-3+3)\hat{k}$$Simplifying each component:$$= \hat{i} - 2\hat{j} + 2\hat{k}$$Now, find the magnitude of this vector:$$| \hat{i} - 2\hat{j} + 2\hat{k} | = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{9} = 3$$The required vector must have magnitude $6$ units.Since it is parallel, it is a scalar multiple of $\hat{i} - 2\hat{j} + 2\hat{k}$.The required scale factor is $\frac{6}{3} = 2$.Thus, the required vector is:$$2(\hat{i} - 2\hat{j} + 2\hat{k}) = 2\hat{i} - 4\hat{j} + 4\hat{k}$$This vector has magnitude $6$ and is parallel to $2\bar{a} - \bar{b} + 3\bar{c}$.Answer:Option A: $2\hat{i} - 4\hat{j} + 4\hat{k}$.