Three Dimension...

If the vectors \(\alpha \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{k}}\) and \(\hat{\gamma}+\gamma \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}\) lie on a plane, where \(\alpha, \beta\) and \(y\) are distinct non-negative numbers, then \(y\) is:

If \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) are four vectors such that \(\vec{a}+\vec{b}+\vec{c}\) is collinear with \(\vec{d}\) and \(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}\) is collinear with \(\overrightarrow{\mathrm{a}}\), then \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}\) is:

Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} ,~ \vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}\) be three non-zero vectors such that \(\vec{c}\) is a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). If the angle between \(\vec{a} \) and \( \vec{b}\) is \(\frac{\pi}{6}\) then \(\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|=?\)

Let \(\vec{a}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\), and \(\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\left(\lambda^{2}-1\right) \hat{\mathrm{k}}\), be coplanar vectors. Then the non zero vector \(\vec{a} \times \vec{c}\) is:

Write unit vector in the direction of the sum of vectors:

\(\overrightarrow{ a }=2 \hat{ i }-\hat{ j }+2 \hat{ k }\) and \(\overrightarrow{ b }=-\hat{ i }+\hat{ j }+3 \hat{ k }\)

The vector \(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\) bisects the angle between the vectors \(\vec{\mathrm{c}}\) and \(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}\). Then unit vector in the direction of \(\vec{c}\) is:

If \(\vec{\mathrm{a}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) and \({\vec{\beta}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-4 \hat{\mathrm{k}}\), The sum of two vectors such that one is parallel to \(\vec{\mathrm{a}}\) and other is perpendicular to \(\overrightarrow{\mathrm{a}}\). Then \(\beta\):

Vector triple product \(\mathrm{a} \times(\mathrm{b} \times \mathrm{c})\) of three vectors \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) is given by:

Find \(|\vec{a} \times \vec{b}|\) if \(\vec{a}=4 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(\vec{b}=6 \hat{i}+7 \hat{j}+8 \hat{k}\).

The sine of the angle between vectors \(\vec{a}=2 \hat{i}-6 \hat{j}-3 \hat{k}\) and \(\vec{b}=4 \hat{i}+3 \hat{j}-\hat{k}\) is: