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Chapter 16: Problem 1558

shortest distance between two lines \(\underline{\mathrm{r}}=(4,-1,0)+\mathrm{k}(1,2,-3), \mathrm{k} \in \mathrm{R}\) and \(\underline{\mathrm{r}}=(1,-1,2)+\mathrm{k}(2,4,-5), \mathrm{k} \in \mathrm{R}\) is (A) \((6 / \sqrt{5})\) (B) \((6 / 5)\) (C) \((\sqrt{6} / 5)\) (D) \(\sqrt{(6 / 5)}\)

Short Answer

Expert verified
The shortest distance between the two lines is \(\dfrac{6\sqrt{86}}{86}\).

Step by step solution

01

Identify the direction ratios of the lines

The given lines can be written in the vector form as: Line 1: \(\underline{\mathrm{r}}_1=(4,-1,0)+\mathrm{k_1}(1,2,-3)\) Line 2: \(\underline{\mathrm{r}}_2=(1,-1,2)+\mathrm{k_2}(2,4,-5)\) The direction ratios for Line 1 are given by the vector \(\textbf{a}\): \(\langle 1, 2, -3 \rangle\) and for Line 2 are given by the vector \(\textbf{b}\): \(\langle 2, 4, -5 \rangle\).
02

Calculate the cross product of direction vectors

To find the vector perpendicular to both lines, calculate the cross product of \(\textbf{a}\) and \(\textbf{b}\): \(\textbf{c}=\textbf{a}\times \textbf{b}=\left\langle 1\times(-5)-2\times(-3), 2\times2-1\times(-5), -3\times2-4\times1\right\rangle\) = \(\left\langle 1, 9, -2 \right\rangle\)
03

Calculate the magnitude of the cross product vector

Compute the magnitude of the cross product vector \(\textbf{c}\): \(|\textbf{c}|=\sqrt{1^2+9^2+(-2)^2}=\sqrt{86}\)
04

Calculate the vector connecting the points on the two lines

Choose a point on each line and find the vector connecting these points. Points \((4,-1,0)\) on Line 1 and \((1,-1,2)\) on Line 2 are selected. The vector connecting these points is: \(\textbf{p}=\left\langle 1-4, -1-(-1), 2-0 \right\rangle = \left\langle -3, 0, 2 \right\rangle\)
05

Compute the shortest distance

Calculate the shortest distance between the two lines using the formula: \(d=\frac{\left|\textbf{p}\cdot\textbf{c}\right|}{|\textbf{c}|}\) \(d=\frac{\left|\left\langle -3, 0, 2 \right\rangle\cdot\left\langle 1, 9, -2 \right\rangle\right|}{\sqrt{86}}\) \(d=\frac{\left|-3(1)+0(9)+2(-2) \right|}{\sqrt{86}}\) \(d=\frac{|6|}{\sqrt{86}}\)
06

Simplify the result by rationalizing the denominator

Simplify the expression by rationalizing the denominator (that is, multiplying both the numerator and denominator by a number such that the denominator becomes rational): \(d=\frac{6}{\sqrt{86}} \times \frac{\sqrt{86}}{\sqrt{86}}=\frac{6\sqrt{86}}{86}\) \(d=\frac{6\sqrt{86}}{86}\) None of the given options matches with the calculated value. Therefore, there might be an error in the answer choices given.

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Most popular questions from this chapter

Direction cosine of line \([(4-x) / 7]=[(y+9) / 5]=[(3 z+8) / 2]\) is (A) \(-[21 / \sqrt{(670)}],[15 / \sqrt{(670)}],[2 / \sqrt{(670)}]\) (B) \([21 / \sqrt{(670)}],[15 / \sqrt{(670)}],[2 / \sqrt{(670)}]\) (C) \([21 / \sqrt{(670)}],[(-15) / \sqrt{(670)}],[2 / \sqrt{(670)}]\) (D) \([(-21) / \sqrt{(670)}],[(-15) / \sqrt{(670)}],[(-2) / \sqrt{(670)}]\)

The unit vector which is perpendicular to \((2,-4,3)\) and \((5,0,1)\), is (A) \([\\{4 / \sqrt{(585)\\}},\\{13 / \sqrt{(585)\\},\\{20 / \sqrt{(585})\\}]}\) (B) \([\\{(-4) / \sqrt{(585)\\}},\\{13 / \sqrt{(585)\\}},\\{(-20) / \sqrt{(585)\\}}]\) (C) \([\\{(-4) / \sqrt{(585)\\}},\\{(-13) / \sqrt{(585)\\},\\{20 / \sqrt{(585})\\}]}\) (D) \([\\{(-4) / \sqrt{(585)\\},\\{13 / \sqrt{(585})\\},\\{20 / \sqrt{(585})\\}]}\)

The equation of plane passing through \((1,6,-4)\) and containing \([(x-1) / 2]=[(y-2) /(-3)]=[(z-3) /(-1)]\) is (A) \(25 \mathrm{x}+14 \mathrm{y}+8 \mathrm{z}=77\) (B) \(25 \mathrm{x}+14 \mathrm{y}-8 \mathrm{y}=77\) (C) \(25 \mathrm{x}-14 \mathrm{y}-8 \mathrm{z}=77\) (D) \(25 x+14 y+8 y=-77\)

For vectors \(\underline{a}, \underline{b}, c \underline{c}\) if each vector is perpendicular to the sum of remaining two vectors and \(|\underline{\mathrm{a}}|=3,|\underline{\mathrm{b}}|=4,|\underline{\mathrm{c}}|=5\), then \(|\underline{a}+\underline{b}+\underline{c}|=\) (A) \(2 \sqrt{2}\) (B) \(3 \sqrt{2}\) (C) \(4 \sqrt{2}\) (D) \(5 \sqrt{2}\)

If \(\ell+\mathrm{m}+\mathrm{n}=0, \ell^{2}-\mathrm{m}^{2}+\mathrm{n}^{2}=0\) and if the direction cosine of two lines are the solution of the given equation, then angle between two line is (A) \((\pi / 2)\) (B) \((\pi / 3)\) (C) \((\pi / 4)\) (D) \((\pi / 6)\)
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