Straight Line in Space

### Find the shortest distance between the given lines.

Answer Given equations: r¯=(ı^+ȷ^)+λ(2ı^-ȷ^+k^) \overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath})+\lambda(2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}})...

### Find the shortest distance between the given lines.

Answer Given equations: r¯=(ı^+ȷ^)+λ(2ı^-ȷ^+k^) \overline{\mathrm{r}}=(\hat{\imath}+\hat{\jmath})+\lambda(2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}})...

### If A(1, 2, 3), B(4, 5, 7), C(-4, 3, -6) and D(2, 9, 2) are four given points then find the angle between the lines AB and CD.

Answer Given - A = (1,2,3) B = (4,5,7) C = (-4,3,-6) D = (2,9,2) Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by ((a’-a),(b’-b),(c’-c))...

### Show that the lines x = – y = 2z and x + 2 = 2y – 1 = – z + 1 are perpendicular to each other.

To prove – The lines are perpendicular to each other Direction ratios of L1 = (2,-2,1) Direction ratios of L2 = (2,1,-2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be...

### prove that

To find – The value of λ Direction ratios of L1 = (-3,2λ,2) Direction ratios of L2 = (3λ,1,-5) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of the second, then...

To prove – The lines are perpendicular to each other Direction ratios of L1 = (2,-3,4) Direction ratios of L2 = (2,4,2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (-3,-2,0) Direction ratios of L2 = (1,-3,2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (1,0,-1) Direction ratios of L2 = (3,4,5) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (2,1,-3) Direction ratios of L2 = (3,2,-1) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (4,3,5) Direction ratios of L2 = (1,-1,1) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines

To find – Angle between the two pair of lines Direction ratios of L1 = (1,1,2) Direction ratios of L2 = (3,5,4) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines:

To find – Angle between the two pair of lines Direction ratios of L1 = (2,-2,1) Direction ratios of L2 = (1,2,-2) Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of...

### Find the angle between each of the following pairs of lines

Tip – If (a,b,c) be the direction ratios of the first line and (a’,b’,c’) be that of the second, then the angle between these pair of lines is given by

### The position vectors of three points A, B and C are

It can thus be written as: A = (-4,2,-3) B = (1,3,-2) C = (-9,1,-4) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line...

### Find the values of λ and μ so that the points A(-1, 4, -2), B(λ, μ 1) and C(0, 2, -1) are collinear.

Answer Given - A = (-1,4,-2) B = (λ,μ,1) C = (0,2,-1) To find – The value of λ and μ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios...

### Find the values of λ and μ so that the points A(3, 2, -4), B(9, 8, -10) and C(λ, μ -6) are collinear.

Answer Given - A = (3,2,-4) B = (9,8,-10) C = (λ,μ,-6) To find – The value of λ and μ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction...

### Find the value of λ for which the points A(2, 5, 1), B(1, 2, -1) and C(3, λ, 3) are collinear.

Answer Given - A = (2,5,1) B = (1,2,-1) C = (3,λ,3) To find – The value of λ so that A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the...

### Show that the points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.

Answer Given - A = (2,3,-4) B = (1,-2,3) C = (3,8,-11) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by...

### Show that the points A(2, 1, 3), B(5, 0, 5) and C(-4, 3, -1) are collinear.

Answer Given - A = (2,1,3) B = (5,0,5) C = (-4,3,-1) To prove – A, B and C are collinear Formula to be used – If P = (a,b,c) and Q = (a’,b’,c’),then the direction ratios of the line PQ is given by...

### Find the image of the point (2, -1, 5) in the line

Find the image of the point $(2,-1,5)$ in the line r→=(11i^-2j^-8k^)+λ(10i^-4j^-11k^) \overrightarrow{\mathrm{r}}=(11 \hat{i}-2 \hat{j}-8 \hat{\mathrm{k}})+\lambda(10 \hat{i}-4...

### Find the image of the point (0, 2, 3) in the line

Find the image of the point $(0,2,3)$ in the line $\frac{\mathrm{x}+3}{5}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}+4}{3}$. Answer Given: Equation of line is...

### Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).

Find the coordinates of the foot of the perpendicular drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$ Answer Given: perpendicular drawn from point...

### Find the coordinates of the foot of the perpendicular drawn from the point A(1, 2, 1) to the line joining the points B(1, 4, 6) and C(5, 4, 4).

Find the coordinates of the foot of the perpendicular drawn from the point $A(1,2,1)$ to the line joining the points $B(1,4,6)$ and $C(5,4,4)$. Answer Given: perpendicular drawn from point...

### Find the vector equation of a line passing through the point having the position vector

Find the vector equation of a line passing through the point having the position vector $(\hat{i}+2 \hat{j}-3 \hat{k})$ and parallel to the line joining the points with position vectors...

### Find the vector equation of a line passing through the point A(3, -2, 1) and parallel to the line joining the points B(-2, 4, 2) and C(2, 3, 3). Also, find the Cartesian equations of the line.

Find the vector equation of a line passing through the point $A(3,-2,1)$ and parallel to the line joining the points $\mathrm{B}(-2,4,2)$ and $\mathrm{C}(2,3,3) .$ Also, find the Cartesian equations...

### Find the vector and Cartesian equations of the line joining the points whose position vectors are

Find the vector and Cartesian equations of the line joining the points whose position vectors are $(\hat{i}-2 \hat{j}+\hat{k})$ and $(\hat{i}+3 \hat{j}-2 \hat{k})$ Answer Given: line passes through...

### Find the vector and Cartesian equations of the line passing through the points A(2, -3, 0) and B(-2, 4, 3).

Find the vector and Cartesian equations of the line passing through the points $A(2,-3,0)$ and $B(-2,4,$, 3). Answer Given: line passes through the points $(2,-3,0)$ and $(-2,4,3)$ To find: equation...

### Find the vector and Cartesian equations of the line passing through the points A(3, 4, -6) and B(5, -2, 7).

Find the equations of the line passing through the point $(1,-2,3)$ and parallel to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-2}{-4}=\frac{\mathrm{Z}+7}{5}$. Also find the vector form of...

### Find the coordinates of the foot of the perpendicular drawn from the point (1, 2, 3) to the line

Find the coordinates of the foot of the perpendicular drawn from the point $(1,2,3)$ to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Also, find the length of the...

### Show that the lines and do not intersect each other.

Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{3}=z$ and $\frac{\mathrm{x}+1}{5}=\frac{\mathrm{y}-2}{1}, Z=2$ do not intersect each other. Answer Given: The equations of the two...

### Show that the lines and intersect each other. Also, find the point of their intersection.

Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{Z}-3}{4}$ and $\frac{\mathrm{x}-4}{5}=\frac{\mathrm{y}-1}{2}=\mathrm{Z}$ intersect each other. Also, find the point...

### Prove that the lines

Prove that the lines $\frac{\mathrm{x}-4}{1}=\frac{\mathrm{y}+3}{4}=\frac{\mathrm{z}+1}{7}$ and $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-3}=\frac{\mathrm{z}+10}{8}$ intersect each other and find...

### Find the Cartesian and vector equations of the line passing through the point (1, 2, -4) and

Find the Cartesian and vector equations of the line passing through the point $(1,2,-4)$ and perpendicular to each of the lines...

### Find the equations of the line passing through the point (-1, 3, -2) and perpendicular to each of the lines and

Find the equations of the line passing through the point $(-1,3,-2)$ and perpendicular to each of the lines $\frac{\mathrm{X}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{Z}}{3}$ and...

### Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line

Find the Cartesian and vector equations of a line which passes through the point $(1,2,3)$ and is parallel to the line $\frac{-\mathrm{X}-2}{1}=\frac{\mathrm{y}+3}{7}=\frac{2 Z-6}{3}$ Answer Given:...

### Find the equations of the line passing through the point (1, -2, 3) and parallel to the line Also find the vector form of this equation so obtained.

Find the equations of the line passing through the point $(1,-2,3)$ and parallel to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-2}{-4}=\frac{\mathrm{Z}+7}{5}$. Also find the vector form of...

### Find the Cartesian equations of the line which passes through the point (1, 3, -2) and is parallel to the line Also, find the vector form of the equations so obtained.

Find the Cartesian equations of the line which passes through the point $(1,3,-2)$ and is parallel to the line given by $\frac{\mathrm{x}+1}{3}=\frac{\mathrm{y}-4}{5}=\frac{\mathrm{z}+3}{-6}$. Also,...

### The Cartesian equations of a line are 3x + 1 = 6y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

The Cartesian equations of a line are $3 x+1=6 y-2=1-z$. Find the fixed point through which it passes, its direction ratios and also its vector equation. Answer Given: Cartesian equation of line are...

### The Cartesian equations of a line Find the vector equation of the line.

The Cartesian equations of a line are $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{Z}-6}{4}$. Find the vector equation of the line. Answer Given: Cartesian equation of line...

A line is drawn in the direction of $(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and it passes through a point with position vector $(2 \hat{i}-\hat{j}-4 \hat{k}) .$ Find the equations...