List of top Mathematics Questions on Equation of a Line in Space

A unit vector parallel to the straight line $\vec{r} = -(5 + 4s)\hat{i} + (7 - 2s)\hat{j} + (3 + 4s)\hat{k}$, where $s$ is the parameter of the line, is

If the equation of the straight line passing through the point \((a,1,3)\) and parallel to the vector \(\frac{2}{3}\hat{i} + \frac{3}{2}\hat{j} + \hat{k}\) is \(\frac{3x+6}{b} = \frac{2y-2}{3} = \frac{z-3}{1}\), then the value of \(a+b\) is equal to

The lines \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \) and \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \) intersect at a point, where \( \vec{a} = \hat{i} + \hat{j} \) and \( \vec{b} = \hat{i} - \hat{k} \). Find the point of intersection.

The vector form of the straight line $\frac{x-2}{1}=\frac{y-1}{-1}=\frac{z-1}{-2}$ is

The equation of line which is parallel to $\frac{2-x}{-3}=\frac{y-2}{2}=\frac{z-4}{1}$ and passing through the point $(1,1,1)$, is

Consider the straight line $\vec{r} = (5\hat{i} + 2\hat{j} - 3\hat{k}) + t(4\hat{i} + 6\hat{j} - 7\hat{k}), \; t \in \mathbb{R}$. Which one of the following points is a point on the straight line?

A straight line passes through the point whose position vector is $\hat{k}$. The straight line also passes through the point of intersection of the lines $\vec{r} = \hat{j} + \lambda \hat{i}, \lambda \in \mathbb{R}$ and $\vec{r} = \hat{i} + s\hat{j}, s \in \mathbb{R}$. Then the equation of the straight line is:

The equation of a line passing through $(-1,2,-4)$ and parallel to the straight line $\dfrac{-x-1}{4} = \dfrac{2y+1}{-1} = \dfrac{-z+4}{3}$, is:

Identify the co-ordinates of the point where the line joining \( (1,1,1) \) and \( (2,2,2) \) intersects the plane \( x+y+z=9 \).

The line passing through the point \( (5, 1, a) \) and \( (3, b, 1) \) crosses the yz-plane at \( (0, \frac{17}{2}, \frac{-13}{2}) \), then the value of \( 2a + 3b \) is:

Which one of the following is a vector parallel to the straight line $\vec{r}=(\hat{i}-11\hat{j}+101\hat{k})+\lambda(3\hat{i}-5\hat{j}+2\hat{k}),\lambda\in\mathbb{R}$? ________.

The equation of the line passing through (0, 0, 1) and (1, 1, 0) is ________.

The point of intersection of the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-11}{4}$ and $\frac{x-3}{1}=\frac{y-\frac{9}{2}}{2}=\frac{z}{1}$ is ________.

If the line $\frac{x-3}{2} = \frac{y+5}{1} = \frac{z+2}{2}$ lies in the plane $\alpha x + 3y - z + \beta = 0$, then values of $\alpha$ and $\beta$ respectively are \dots}

The altitude through vertex A of $\triangle ABC$ with position vectors of points A, B, C as $\vec{a}, \vec{b}, \vec{c}$ respectively is ______.

A triangle ABC is formed by A(1, -1, 0), B(3, 5, 3), C(-11, -5, 6). The equation of the internal angle bisector of angle A is ______.

The line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z}{1}$ intersects the XY and YZ planes at A and B. The line through A and B is}

The lines $\frac{6x-6}{18}=\frac{y+1}{3}=\frac{z-1}{5}$ and $\frac{3x+6}{12}=\frac{y-1}{3}=\frac{z+1}{2}$ are...

The line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z}{1}$ intersects the XY and YZ planes at A and B. The line through A and B is}

The lines $\frac{6x-6}{18}=\frac{y+1}{3}=\frac{z-1}{5}$ and $\frac{3x+6}{12}=\frac{y-1}{3}=\frac{z+1}{2}$ are...

The equation of a line passing through the point $(-1, 2, 3)$ and perpendicular to the lines $\frac{x}{2} = \frac{y-1}{-3} = \frac{z+2}{-2}$ and $\frac{x+3}{-1} = \frac{y+3}{2} = \frac{z-1}{3}$ is

The number of solutions of \(16^{\sin^2 x} + 16^{\cos^2 x} = 10\) in \(0 \le x \le 2\pi\) are

If the point \( (3,6,k) \) lies on the line \( \dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3} \), then the value of \( k \) is

If a point \( P \) with \( x \)-coordinate \( 7 \) lies on the line joining the points \( A(1,2,3) \) and \( B(4,6,8) \), then the coordinates of the point \( P \) are

The equation of the straight line joining the points \((1,2,3)\) and \((3,4,k)\) is \(\frac{x-3}{1}=\frac{y-4}{1}=\frac{z-k}{5}\). Then the value of \(k\) is