Q 16. - Calculus | StudyQuestionHub

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Q 16.

Question

Let $x = a t + x 0, y = b t + y_{0}, z = c t + z_{0}$ be parametric equations for a line $L$ in $R_{3}$ . If x 0, y0, and z 0 are all nonzero, give conditions on a, b, and c so that

(a) L intersects all three coordinate planes.

(b) L intersects the $x y$ and $y z -$ planes, but not the

$x z -$ plane.

(c) $L$ intersects exactly one of three coordinate planes.

Step-by-Step Solution

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Answer

Part (a) $x_{0} = - a t, y_{0} = - b t, y_{0} = - c t$

Part (b) $x = a t + x_{0}, y = b t + y_{0}, 0 = c t + z_{0}$ and $0 = a t + x_{0}, y = b t + y_{0}, z = c t + z_{0}$

Part (c) $a t + x_{0}, y = b t + y_{0}, c t + z_{0} = 0$

1Part (a) Step 1: Given information

$x = a t + x_{0}, y = b t + y_{0}, z = c t + z_{0}$

2Part (a) Step 2: Calculation

Consider $x = a t + x_{0}, y = b t + y_{0}, z = c t + z_{0}$ be parametric equations for a line $L$ in $ℝ^{3}$

Consider when $L$ intersects all the three coordinate planes.

Consider what happens when $mathcalL$ crosses all three coordinate planes. Only at the origin is it possible to intersect all three points.

The values of $x, y, z$ become 0 when the line touches all three coordinate planes. Then the line equation in parametric form becomes $0 = x_{0} + a t, 0 = y_{0} + b t, 0 = z_{0} + c t$

$x_{0} = - a t, y_{0} = - b t, y_{0} = - c t$

Therefore, the answer is $x_{0} = - a t, y_{0} = - b t, y_{0} = - c t$

3Part (b) Step 1: Calculation

Consider the line which intersects $x y, y z$ coordinate planes.

The value of the $z$ co-ordinate becomes zero when the line touches the $x y -$ plane.

$x = a t + x_{0}, y = b t + y_{0}, 0 = c t + z_{0}$

Therefore, the answer is $x = a t + x_{0}, y = b t + y_{0}, 0 = c t + z_{0}$

The value of the $x$ co-ordinate becomes zero when the line intersects the $y z -$ plane.

$0 = a t + x_{0}, y = b t + y_{0}, z = c t + z_{0}$

Therefore, the answer is $0 = a t + x_{0}, y = b t + y_{0}, z = c t + z_{0}$

4Part (c) Step 1: Calculation

If the line intersects $x y -$ plane.

The value of $z$ becomes 0 when a line touches the $x y -$ plane.

Then the equation in parametric form is $x = a t + x_{0}, y = b t + y_{0}, 0 = c t + z_{0}$

Therefore the answer is $a t + x_{0}, y = b t + y_{0}, c t + z_{0} = 0$

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