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Chapter 12: Problem 9

Find equations of the following lines. The line through (0,0,1) in the direction of the vector \(\mathbf{v}=\langle 4,7,0\rangle\)

Short Answer

Expert verified
Question: Find the parametric equations of the line that passes through the point (0,0,1) and moves in the direction of the vector \( \mathbf{v} =\langle 4, 7, 0 \rangle \). Answer: The parametric equations of the line are \( x = 4t, y = 7t, z = 1\).

Step by step solution

01

Identify the given point and direction of the line

The given point is \(P(0,0,1)\) and the direction vector of the line is \(\mathbf{v} = \langle 4, 7, 0 \rangle\).
02

Write the parametric equations of the line

The parametric equations of the line are derived from the given point and direction vector: \(x = x_0 + at\) \(y = y_0 + bt\) \(z = z_0 + ct\) where \((x_0, y_0, z_0)\) is the given point, \(\mathbf{v} = \langle a, b, c \rangle\) is the direction vector, and t is the parameter.
03

Plug in the given point and direction vector into the parametric equations

Using the given point and direction vector, we plug the values for \((x_0, y_0, z_0)\) and \((a, b, c)\) into the parametric equations: \(x = 0 + 4t\) \(y = 0 + 7t\) \(z = 1 + 0t\)
04

Simplify the parametric equations

Simplify the parametric equations to get the final result: \(x = 4t\) \(y = 7t\) \(z = 1\) These parametric equations represent the line that passes through the point (0,0,1) and moves in the direction of the vector \(\mathbf{v} = \langle 4, 7, 0 \rangle\).

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