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List of top Mathematics Questions on Vectors asked in BITSAT

Let vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be such that $$ \mathbf{a} = \hat{i} + 2\hat{j} - \hat{k}, \quad \mathbf{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \mathbf{c} = \hat{i} + \hat{j} + \hat{k} $$ Then the volume of the parallelepiped formed by these vectors is:
If \( \vec{p} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \vec{q} = \hat{i} + 4\hat{j} - \hat{k} \), and \( \vec{r} = 2\hat{i} - 3\hat{j} + 5\hat{k} \), find \( \vec{p} \cdot (\vec{q} \times \vec{r}) \).
If \( \vec{a} = \hat{i} + 2\hat{j} - \hat{k} \), \( \vec{b} = 2\hat{i} - \hat{j} + 3\hat{k} \), and \( \vec{c} = -\hat{i} + 3\hat{j} + 2\hat{k} \), find \( \vec{a} \cdot (\vec{b} \times \vec{c}) \).
The angle between the lines whose direction cosines are given by the equations \( 3l + m + 5n = 0 \) and \( 6m - 2n + 5l = 0 \) is:
The magnitude of projection of the line joining \( (3,4,5) \) and \( (4,6,3) \) on the line joining \( (-1,2,4) \) and \( (1,0,5) \) is:
Let \( ABC \) be a triangle and \( \vec{a}, \vec{b}, \vec{c} \) be the position vectors of \( A, B, C \) respectively. Let \( D \) divide \( BC \) in the ratio \( 3:1 \) internally and \( E \) divide \( AD \) in the ratio \( 4:1 \) internally. Let \( BE \) meet \( AC \) in \( F \). If \( E \) divides \( BF \) in the ratio \( 3:2 \) internally then the position vector of \( F \) is:
Let \( \mathbf{a} = \hat{i} - \hat{k}, \mathbf{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \mathbf{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k} \). Then, \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] \) depends on:}
If \( a, c, b \) are in GP, then the area of the triangle formed by the lines \( ax + by + c = 0 \) with the coordinate axes is equal to: