Let a = i + 4 j +2 k , b = 3 i - 2 j +7 k and c = 2 i - j + 4 k F… - Vidyadip

Question

Let $\overrightarrow{\text{a}} = \hat{\text{i}} + 4\hat{\text{j}} +2\hat{\text{k}}, \overrightarrow{\text{b}} = 3\hat{\text{i}} - 2\hat{\text{j}} +7\hat{\text{k}}$ and $\overrightarrow{\text{c}} = 2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ Find a vector $\overrightarrow{\text{d}}$ which is perpendicular to both $\overrightarrow{\text{a}} \text{and} \overrightarrow{\text{b}}\text{and} \overrightarrow{\text{c}} . \overrightarrow{\text{d}} = 27.$

✓

Answer

$\text{Writing} \overrightarrow{\text{d}} = \lambda\bigg(\overrightarrow{\text{a}}\times\overrightarrow{\text{b}}\bigg)$
$= \lambda \begin{vmatrix} \hat{\text{i}} & \hat{\text{j}} & \hat{\text{k}} \\ 1 & 4 & 2 \\ 3 & -2 & 7 \end{vmatrix} $
$ = \lambda \bigg(32 \hat{\text{i}} - \hat{\text{j}} - 14\hat{\text{k}}\bigg)\dots\dots\dots\dots\text{(1)}$
$\overrightarrow{\text{c}}. \overrightarrow{\text{d}} = 27$
$\bigg(2\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}\bigg).\lambda\bigg(32\hat{\text{i}} - \hat{\text{j}} + 14\hat{\text{k}}\bigg) = 27$
$9\lambda = 27$
$\lambda = 3$
$\therefore\overrightarrow{\text{d}} = \bigg(96\hat{\text{i}} - \hat{\text{3j}} + 42\hat{\text{k}}\bigg)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the area of the region in the first quadrant enclosed by x-axis, the line $\text{y}=\sqrt{3\text{x}}$ and the circle $x^2+ y^2 = 16$.

Find perpendicular distance of the point $(1, 0, 0)$ from the line $\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8}$ .Also, find the coordinates of the foot of the perpendicular and the equation of the perpendicular.

Solve the following determinant equations:
$\begin{vmatrix}15-2\text{x}&11-3\text{x}&7-\text{x}\\11&17&14\\10&16&13\end{vmatrix}=0$

Find the vector and cartesian forms of the plane passing through the point (1, 2, -4) and parallel to the lines $\vec{\text{r}}=(\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}})+\lambda(2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}})$ and $\vec{\text{r}}=(\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}})+\mu(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}).$ Also, find the distance of the point (9, -8, -10) from the plane thus obtained.

Find an equation for the set all points that are equidistant from the planes $3x - 4y + 12z = 6$ and $4x + 3z = 7$

If $\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1+\cos\theta)$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$

Find the general solution of $x \frac{d y}{d x}+2 y=x^{2} \log x$

A and B are partners sharing profits and losses in the ratio 3 : 4 respectively. They admit C as a new partner, the new profit sharing ratio being 2 : 2 : 3 between A, B and C respectively. C pays Rs. 12,000 as premium for goodwill. Find the amount of premium shared by A and B.

If y(x) is a solution of $\Big(\frac{2+\sin\text{x}}{1+\text{y}}\Big)\frac{\text{dy}}{\text{dx}}=-\cos\text{x}$ and y(0) = 1, then find the value of $\text{y}\big(\frac{\pi}{2}\big).$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$