List of top Mathematics Questions on Applications of Vectors asked in CBSE Class XII

If \( A, B \) and \( C \) be three non-collinear points such that \( \vec{AB} = \hat{i} + 2\hat{j} - \hat{k} \) and \( \vec{AC} = 2\hat{i} - 3\hat{j} \), then find the area of \( \triangle ABC \).

If \(A, B\) and \(C\) be three non-collinear points such that \(\vec{AB} = \hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{AC} = 2\hat{i} - 3\hat{j}\), then find the area of \(\Delta ABC\).

Let two rods placed on the ground be represented by vectors \( 4\hat{i} - \hat{j} + 3\hat{k} \) and \( -2\hat{i} + \hat{j} - 2\hat{k} \). Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

Three points \(A(0, 1, 1)\), \(B(2, 0, -1)\) and \(C(1, 0, 3)\) form \(\Delta ABC\). The \(ar (\Delta ABC)\) is :

If \( \vec{AB} = \hat{j} + \hat{k} \) and \( \vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k} \) represent the two vectors along the sides \( AB \) and \( AC \) of \( \Delta ABC \), prove that the median \( \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \), where \( D \) is midpoint of \( BC \). Hence, find the length of median \( AD \).

Using vectors, find the area of \( \Delta ABC \) with vertices \( A(1, 2, 3) \), \( B(2, -1, 4) \) and \( C(4, 5, -1) \).

Vectors \( \vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{k} \) represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

If $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(2\hat{i} + \hat{j} - \hat{k})$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.

If \(\vec{AB} = \hat{j} + \hat{k}\) and \(\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}\) represent two vectors along the sides \(AB\) and \(AC\) of \(\Delta ABC\), prove that the median vector satisfies \(\vec{AB} + \vec{AC} = 2\vec{AD}\), where \(D\) is the midpoint of \(BC\). Hence, find the length of the median \(AD\).