Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$ $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})+\mu(-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})$ Solution:...
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Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ $\overrightarrow{\mathrm{r}}=(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(-4 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$ $\overrightarrow{\mathrm{r}}=(-3 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ Solution:...
Read More →Find the shortest distance between the given lines.
Question: Find the shortest distance between the given lines. $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+\lambda(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$ Solution:...
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Question: 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 $A B$ and $C D$. Solution:...
Read More →Find the angle between two lines whose direction ratios are
Question: Find the angle between two lines whose direction ratios are i. $2,1,2$ and $4,8,1$ ii. $5,-12,13$ and $-3,4,5$ iii. $1,1,2$ and $(\sqrt{3}-1),(-\sqrt{3}-1), 4$ iv. $a, b, c$ and $(b-c),(c-a),(a-b)$ Solution:...
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Question: Show that the lines $x=-y=2 z$ and $x+2=2 y-1=-z+1$ are perpendicular to each other. HINT: The given lines are $\frac{x}{2}=\frac{y}{-2}=\frac{z}{1}$ and $\frac{x+2}{1}=\frac{y-1 / 2}{1}=\frac{z-1}{-2}$. Solution:...
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Question: If the lines $\frac{x-1}{-3}=\frac{y-2}{2 \lambda}=\frac{z-3}{2}$ and $\frac{x-1}{3 \lambda}=\frac{y-1}{1}=\frac{6-z}{5}$ are perpendicular to each other then find the value of $\lambda$. Solution:...
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Question: Show that the lines $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}+1}{-3}=\frac{\mathrm{z}-2}{4}$ and $\frac{\mathrm{x}+2}{2}=\frac{\mathrm{y}-4}{4}=\frac{\mathrm{z}+5}{2}$ are perpendicular to each other. Solution:...
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Question: Find the angle between each of the following pairs of lines: $\frac{5-x}{3}=\frac{y+3}{-2}, z=5$ and $\frac{x-1}{1}=\frac{1-y}{3}=\frac{z-5}{2}$ Solution:...
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Question: Find the angle between each of the following pairs of lines: $\frac{x}{1}=\frac{z}{-1}, y=0$ and $\frac{x}{3}=\frac{y}{4}=\frac{z}{5}$ Solution:...
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Question: Find the angle between each of the following pairs of lines: $\frac{3-x}{-2}=\frac{y+5}{1}=\frac{1-z}{3}$ and $\frac{x}{3}=\frac{1-y}{-2}=\frac{z+2}{-1}$ Solution:...
Read More →Find the angle between each of the following pairs of lines:
Question: Find the angle between each of the following pairs of lines: $\frac{\mathrm{x}-4}{4}=\frac{\mathrm{y}+1}{4}=\frac{\mathrm{z}-6}{5}$ and $\frac{\mathrm{x}-5}{1}=\frac{2 \mathrm{y}+5}{-2}=\frac{\mathrm{z}-3}{1}$ Solution:...
Read More →Find the angle between each of the following pairs of lines:
Question: Find the angle between each of the following pairs of lines: $\frac{x-1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$ and $\frac{x+3}{3}=\frac{y-2}{5}=\frac{z+5}{4}$ Solution:...
Read More →Find the angle between each of the following pairs of lines:
Question: Find the angle between each of the following pairs of lines: $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}})+\lambda(2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=3 \hat{\mathrm{k}}+\mu(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ Solution:...
Read More →Find the angle between each of the following pairs of lines:
Question: Find the angle between each of the following pairs of lines: $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+3 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=5 \hat{\mathrm{i}}+\mu(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ Solution:...
Read More →Find the angle between each of the following pairs of lines:
Question: Find the angle between each of the following pairs of lines: $\overrightarrow{\mathrm{r}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-5 \hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-4 \hat{\mathrm{k}})$ Solution:...
Read More →The position vectors of three points A, B and C are
Question: The position vectors of three points $A, B$ and $C$ are $(-4 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}),(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ and $(-9 \hat{i}+\hat{j}-4 \hat{k})$ respectively. show that the points $A, B$ and $C$ are collinear. Solution:...
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Question: Find the values of $\lambda$ and $\mu$ so that the points $A(-1,4,-2), B(\lambda, \mu 1)$ and $C(0,2,-1)$ are collinear. Solution:...
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Question: Find the values of $\lambda$ and $\mu$ so that the points $A(3,2,-4), B(9,8,-10)$ and $C(\lambda, \mu-6)$ are collinear. Solution:...
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Question: Find the value of $\lambda$ for which the points $A(2,5,1), B(1,2,-1)$ and $C(3, \lambda, 3)$ are collinear. Solution:...
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Question: Show that the points $A(2,3,-4), B(1,-2,3)$ and $C(3,8,-11)$ are collinear. Solution:...
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Question: Show that the points $\mathrm{A}(2,1,3), \mathrm{B}(5,0,5)$ and $\mathrm{C}(-4,3,-1)$ are collinear. Solution:...
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Question: Find the image of the point $(2,-1,5)$ in the line $\overrightarrow{\mathrm{r}}=(11 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-8 \hat{\mathrm{k}})+\lambda(10 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-11 \hat{\mathrm{k}})$ Solution:...
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