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

If \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) are unit vectors, then prove that \( |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9 \).

Three honey bees were found flying along the vectors \( \vec{a} = 2\hat{i} - 3\hat{j} + \hat{k} \), \( \vec{b} = 4\hat{j} - 2\hat{k} \) and \( \vec{c} = 3\hat{i} + 2\hat{k} \) respectively. Find the value of \( \lambda \) such that the path for \( \vec{a} + \lambda \vec{b} \) is perpendicular to \( \vec{c} \).

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are unit vectors, then prove that \( |\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9 \).

Three honey bees were found flying along the vectors \(\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}\), \(\vec{b} = 4\hat{j} - 2\hat{k}\) and \(\vec{c} = 3\hat{i} + 2\hat{k}\) respectively.
Find the value of \(\lambda\) such that the path for \(\vec{a} + \lambda\vec{b}\) is perpendicular to \(\vec{c}\).

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

For any two vectors \( \vec{a} \) and \( \vec{b} \), which of the following statements is always true ?

The value of \( p \) for which vectors \( \hat{i} + 2\hat{j} + 3\hat{k} \) and \( 2\hat{i} - p\hat{j} + \hat{k} \) are perpendicular to each other is