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Chapter 16: Problem 1554

Lines \([\\{(\mathrm{k}+3),-\mathrm{k}-1, \mathrm{k}+1\\} / \mathrm{k} \in \mathrm{R}\\},\\{(2 \mathrm{k}, 0,3 \mathrm{k}-3) / \mathrm{k} \in \mathrm{R}\\}\) are (A) parallel (B) Intersecting (C) coincident (D) skew

Short Answer

Expert verified
The correct answer is (D) skew. The lines are not parallel, intersecting, or coincident, and hence, they are skew lines.

Step by step solution

01

Identify the Direction Vectors

For the first line, the direction vector \(\vec{d_1}\) is given by (1, -1, 1). For the second line, the direction vector \(\vec{d_2}\) is given by (2, 0, 3).
02

Check if the Lines are Parallel

Two lines are parallel if their direction vectors are proportional. That is, \(\vec{d_1} = k\vec{d_2}\) for some scalar \(k\). However, here, the direction vectors (1, -1, 1) and (2, 0, 3) are not proportional because there is no constant \(k\) that can multiply \(\vec{d_1}\) to give \(\vec{d_2}\). Thus, the lines are not parallel.
03

Check if the Lines Intersect

For the two lines to intersect, there must be a common point between them. Let's equate the vector equations of the lines. For some values of \(k\) in each equation, the lines should give a common point. If we equate the vectors, we get the system of equations: (1) \(k_1 + 3 = 2k_2\) (2) \(-k_1 - 1 = 0\) (3) \(k_1 + 1 = 3k_2 - 3\) On solving these equations, we do not arrive at a consistent solution. Thus, the lines do not intersect.
04

Check if the Lines are Coincident

The lines are coincident if they are effectively the 'same' line. This condition arises when the direction vectors are the same and the point vectors on the line can be set equal by adjusting the values of \(k\). Since the direction vectors are not the same in this case, the lines cannot be coincident.
05

Conclude the Relationship between Lines

Based on the above analysis, it is clear that these lines are not parallel, intersecting, or coincident. Therefore, the only remaining option is that these lines are skew. As such, the correct answer is (D) skew.

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Most popular questions from this chapter

If angle between two units vectors \(\underline{a}\) and \(\underline{b}\) is \(\theta\), then \(\sin (\theta / 2)=\) (A) \(|\underline{a}+\underline{b}|\) (B) \((1 / 2)|\underline{a}-\underline{b}|\) (C) \(|\underline{a}-\underline{b}|\) (D) \((1 / 2)|\underline{a}+\underline{b}|\)

\(\underline{\mathrm{x}}=(2,-6,3), \mathrm{y}=(1,2,-2)\) and \(\underline{\mathrm{x}}^{\wedge} \mathrm{y}=\theta\), then \(\sin \theta=\) (A) \([21 / \sqrt{(185)}]\) (B) \(-[\sqrt{(185) / 21]}\) (C) \(-[21 / \sqrt{(185)}]\) (D) \([\sqrt{(185) / 21]}\)

For vector \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) if each vector forms an angle \((\pi / 3)\) with remaining two vectors and \(|\underline{a}|=1,|\underline{b}|=2,|\underline{c}|=3\), then \(|\underline{a}+\underline{b}+\underline{c}|=\) (A) \(\sqrt{17}\) (B) 0 (C) 5 (D) \(\sqrt{5}\)

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) \((\pi / 6)\)

If the vertices of quadrilateral are \((1,1,1),(-2,4,1)\) \((-1,5,5),(2,2,5)\) then it is (A) rectangle (B) square (C) parallelogram (D) rhombus
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