A line passes through (2, –1, 3) and is perpendicular to the line… - Vidyadip

Question

A line passes through (2, –1, 3) and is perpendicular to the lines$\overrightarrow{\text{r}}= (\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}) + \lambda(2 \hat{\text{i}} - 2\hat{\text{j}} + \hat{\text{k}})\text{ and }\overrightarrow{\text{r}} = (2\hat{\text{i}} -\hat{\text{j}} - 3\hat{\text{k}}) + \mu(\hat{\text{i}} + 2 \hat{\text{j}} + 2\hat{\text{k}}).$Obtain its equation in vector and cartesian form.

✓

Answer

The direction perpendicular to the given lines is given by
$(2\hat{\text{i}} - 2 \hat{\text{j}} +\hat{\text{k}})\times(\hat{\text{i}} + 2\hat{\text{j}} + 2\hat{\text{k}})$
$= \begin{bmatrix} \hat{\text{i}} &\hat{\text{j}}&\hat{\text{k}} \\[0.3em] 2 & -2 & 1 \\[0.3em] 1 & 2& 2 \end{bmatrix} = -6\hat{\text{i}} - 3\hat{\text{j}} + 6\hat{\text{k}}\text{ or }2\hat{\text{i}} +\hat{\text{j}} - 2\hat{\text{k}}$
$\therefore$ Vector equation of required line is
$\overrightarrow{\text{r}} = (2\hat{\text{i}} -\hat{\text{j}} + 3 \hat{\text{k}}) + \lambda(2\hat{\text{i}} + \hat{\text{j}} - 2 \hat{\text{k}})$
and the cartesian form is
$\frac{\text{x} - 2}{2} = \frac{\text{y} + 1 }{1} =\frac{\text{z} - 3}{-2}.$

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 "4 Marks" 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

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_\frac{\pi}{3}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{3}{2}}}\text{ dx}$

$\int\frac{\text{x}+1}{\sqrt{2\text{x}+3}}\text{dx}$

Find the angle between the pairs of lines with direction ratios proportional to

1, 2, -2 and -2, 2, 1

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?

Solve the following differential equations:
$\text{x}\frac{\text{dy}}{\text{dx}}=\text{x + y}$

Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.

If $\text{f}\text{(x)}=\begin{cases}\frac{2^\text{z+2}-16}{4^\text{x}-16}, &\text{if x} \neq 2\\\text{k}, & \text{x} = 2\end{cases}$
is continuous at x = 2, Find k.

Find the equation of the plane passing through the intersection of the planes 2x + 3y - z + 1 = 0 and x + y - 2z + 3 = 0 and perpendicular to the plane 3x - y - 2z - 4 = 0.

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Evaluate the following integrals:
$\int\limits_{0}^{\text{a}}\frac{\text{x}}{\sqrt{\text{a}^2+\text{x}^2}}\text{ dx}$