Relations between Points, Lines and Planes Test
Chapter
Chapter 9
Section
Relations between Points, Lines and Planes Test
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Solutions 7 Videos

a) Determine the point of intersection for the lines having equations \vec{r} = (4, 2, 6) +s(1, ,3 11), x \in \mathbb{R} and \vec{r} = (5, -1, 4) +t(2, 0, 9), t\in \mathbb{R}.

b) Verify that the intersection point of these two lines is on the plane.

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Q1

a) Determine the distance from point A(3, 2, 3) to \pi: 8x -8y + 4z - 7 = 0.

b) Determine the distance between the planes \pi_1: 2x - y -2z - 16 = 0 and \pi_2: 2x - y + 2z + 24 = 0.

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Q2

a) Determine the equation of the line of intersection L between the planes \pi_1: 2x + 3y - z =3 and \pi_2: -x + y + z = 1.

b) Determine the point of intersection between L and xz-plane.

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Q3

Solve the following system of equations:

\displaystyle \begin{array}{lllll} & \phantom{.} x - y + z = 10 \\ & \phantom{.} 2x + 3y -2z = -21 \\ & \phantom{.} \frac{1}{2}x + \frac{2}{5}y + \frac{1}{4}z = -\frac{1}{2} \end{array}

Explain what your solution means geometrically.

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Q4

Solve the following system of equations:

\displaystyle \begin{array}{lllll} & \phantom{.} x - y + z = -1 \\ & \phantom{.} 2x + 2y -z = 0 \\ & \phantom{.} x -5y +4z = -3 \end{array}

Explain what your solution means geometrically.

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Q5

The three planes x + y + z = 0, x + 2y + 2z = 1, and 2x - y + mz = n intersect in a line.

a) Determine the values of m and n for which this is true.

b) What is the equation of the line?

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Q6

Determine the distance between the skew lines with equations

\displaystyle \begin{array}{lllll} & \phantom{.} L_1: & \vec{r} = (-1, -3, 0) +s(1, 1, 1), s \in \mathbb{R}\\ & \phantom{.} L_2: & \vec{r} = (-5, 5, -8) +t(1, 2, 5), t \in \mathbb{R} \end{array}

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Q7