If three vectors 2hat{i}-hat{j}-hat{k},hat{i} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $\vec{a} = 2\hat{i} - \hat{j} - \hat{k}$,$\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k}$.Since

Vedclass · Accepted Answer

$-8$

Vedclass · Answer

(D) Let $\vec{a} = 2\hat{i} - \hat{j} - \hat{k}$,$\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k}$.Since these vectors are coplanar,their scalar triple product must be zero,i.e.,$[\vec{a} \vec{b} \vec{c}] = 0$.This implies the determinant of the components is zero:$\begin{vmatrix} 2 & -1 & -1 \ 1 & 2 & -3 \ 3 & \lambda & 5 \end{vmatrix} = 0$Expanding the determinant along the first row:$2(2(5) - (-3)(\lambda)) - (-1)(1(5) - (-3)(3)) + (-1)(1(\lambda) - 2(3)) = 0$$2(10 + 3\lambda) + 1(5 + 9) - 1(\lambda - 6) = 0$$20 + 6\lambda + 14 - \lambda + 6 = 0$$5\lambda + 40 = 0$$5\lambda = -40$$\lambda = -8$.

If three vectors $2\hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda\hat{j}+5\hat{k}$ are coplanar,then the value of $\lambda$ is

Similar Questions

If $x \cdot a = 0, x \cdot b = 0$ and $x \cdot c = 0$ for some non-zero vector $x$,then the true statement is

If the vectors $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} + 2\hat{j} - 3\hat{k}$,and $3\hat{i} + a\hat{j} + 5\hat{k}$ are coplanar,then the value of $a$ is:

The value of $[\vec{a}-\vec{b} \quad \vec{b}-\vec{c} \quad \vec{c}-\vec{a}]$ is equal to

If $a, b,$ and $c$ are unit coplanar vectors,then the scalar triple product $[a, b, c]$ is equal to:

For non-zero vectors $\vec{a}, \vec{b}, \vec{c}$,the condition $|(\vec{a} \times \vec{b}) \cdot \vec{c}| = |\vec{a}||\vec{b}||\vec{c}|$ holds if and only if:

Vedclass Test Series

Exam Paper Generator

Online Exam Module