List of top Mathematics Questions on Geometry and Vectors

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three vectors such that \( |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = \sqrt{3} \) and \( ( \mathbf{a} + \mathbf{b} - \mathbf{c} )^2 + ( \mathbf{b} + \mathbf{c} - \mathbf{a} )^2 + ( \mathbf{c} + \mathbf{a} - \mathbf{b} )^2 = 36\), then \(|2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2 =\)
Let ABC be an equilateral triangle of side \(a\). M and N are two points on the sides AB and AC respectively such that \(AN = K \cdot AC\) and \(AB = 3 \cdot AM\). If the vectors \(BN\) and \(CM\) are perpendicular, then \(K = \) ?

Find the shortest distance between the skew lines $\vec{r} = (-\hat{i} - 2\hat{j} - 3\hat{k}) + t(3\hat{i} - 2\hat{j} - 2\hat{k})$ and $\vec{r} = (7\hat{i} + 4\hat{k}) + s(\hat{i} - 2\hat{j} + 2\hat{k})$.

In a regular hexagon \( ABCDEF \), if \( \overrightarrow{AB} = \mathbf{a} \) and \( \overrightarrow{BC} = \mathbf{b} \), then find \( \overrightarrow{FA} \).
If \( P_1, P_2, P_3 \) are the lengths of the altitudes drawn from the vertices \( A, B, C \) of \( \triangle ABC \) respectively, then: \[ \cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = ? \]
If \( \mathbf{a} = (2x + y)\mathbf{i} + 3\mathbf{j} + 9\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} + \mathbf{j} - (x - y)\mathbf{k} \) are two collinear vectors, then \( x^3 + 27y^3 = \)
If a point \( C \) divides the line segment joining the points with position vectors \( 2\mathbf{i} - 3\mathbf{j} + 2\mathbf{k} \) and \( 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} \) in the ratio \( 2:3 \), then the distance of \( C \) from the point with position vector \( 2\mathbf{i} - \mathbf{j} + \mathbf{k} \) is
Let \( \mathbf{a} = \mathbf{i} - 2\mathbf{j} \), \( \mathbf{b} = 2\mathbf{j} + 3\mathbf{k} \), \( \mathbf{c} = p\mathbf{i} + q\mathbf{j} \) and \( \mathbf{d} = p\mathbf{j} - q\mathbf{k} \) be four vectors. If \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = 3 \) and \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{d} = 0 \), then \( 3p + q = \)
Let \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{b} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k} \) be two vectors. If \( A_1 \) is the area of the quadrilateral having \( \mathbf{a}, \mathbf{b} \) as its diagonals and \( A_2 \) is the area of the parallelogram having \( \mathbf{a}, \mathbf{b} \) as its adjacent sides, then \( A_1 : A_2 = \)
For some real number \( \lambda \), if the area of the triangle having \( \mathbf{a} = \lambda \mathbf{i} - 3\mathbf{j} + \mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} + \lambda \mathbf{j} - 3\mathbf{k} \) as two of its sides is \( \frac{\sqrt{195}}{2} \), then the number of distinct possible values of \( \lambda \) is