Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector v = <<4, 7, 0>>

Question
Equations
asked 2021-01-19
Equations of lines Find both the parametric and the vector equations of the following lines.
The line through (0, 0, 1) in the direction of the vector \(\displaystyle{v}={\left\langle{4},{7},{0}\right\rangle}\)

Answers (1)

2021-01-20
Step 1
Given:
The line through (0, 0, 1) in the direction of the vector \(\displaystyle{v}={\left\langle{4},{7},{0}\right\rangle}\).
To find:
Parametric and vector equations of the given line.
Step 2
Let,
\(\displaystyle\vec{{{r}_{{0}}}}={\left({0},{0},{1}\right)},\vec{{{d}}}={<}{4},{7},{0}{>}\)
The vector equation of the line is,
\(\displaystyle\vec{{{r}}}=\vec{{{r}_{{0}}}}+{t}\vec{{{d}}}\)
\(\displaystyle\Rightarrow\vec{{{r}}}={<}{0},{0},{1}{>}+{t}{<}{4},{7},{0}{>}\)
\(\displaystyle={<}{0},{0},{1}{>}+{<}{4}{t},{7}{t},{0}{>}\)
\(\displaystyle={<}{0}+{4}{t},{0}+{7}{t},{1}+{0}{>}\)
\(\displaystyle={<}{4}{t},{7}{t},{1}{>}\)
\(\displaystyle\Rightarrow\vec{{{r}}}={<}{4}{t},{7}{t},{0}{>}\)
The parametric equations of a line are,
Result:x = 4t, y = 7t, z = 0
Vector equation of a line is, \(\displaystyle\vec{{{r}}}={<}{4}{t},{7}{t},{0}{>}\)
Parametric equations of a line are,
x=4t, y = 7t, z =0.
0

Relevant Questions

asked 2020-12-07
Equations of lines Find both the parametric and the vector equations of the following lines. The line through (0, 0, 1) in the direction of the vector \(v = ⟨4, 7, 0⟩\)
asked 2021-02-02
Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.) Point \((-1,\ 0,\ 8)\) Parallel to \(v = 3i\ +\ 4j\ -\ 8k\) The given point is \((−1,\ 0,\ 8)\ \text{and the vector or line is}\ v = 3i\ +\ 4j\ −\ 8k.\) (a) parametric equations (b) symmetric equations
asked 2020-11-22
Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values.
The line segment starting at P(-1, -3) and ending at Q(6, -16)
asked 2021-02-08
Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
\(\displaystyle{x}=-{7}{\cos{{2}}}{t},{y}=-{7}{\sin{{2}}}{t},{0}\le{t}\le\pi\)
asked 2020-11-14
a) Find parametric equations for the line. (Use the parameter t.)
The line of intersection of the planes
x + y + z = 2 and x + z = 0
(x(t), y(t), z(t)) =
b) Find the symmetric equations.
asked 2021-02-12
Find parametric equations for the lines in The line through the origin parallel to the vector \(2j + k\)
asked 2020-11-02
Express the equations of the lines
r · (i −j) + 7 = 0,r · (i + 3j) − 5 = 0 in parametric forms and hence find the position
vector of their point of intersection.
asked 2020-10-27
Which of the following equations have the same solution set? Give reasons for your answers that do not depend on solving the equations.
l.\(x-5=3x+7\)
ll.\(3x-6=7x+8\)
lll.\(15x-9=6x+24\)
lV.\(6x-16=14x+12\)
V.\(9x+21=3x-15\)
Vl.\(-0.05+\frac{x}{100}=3\frac{x}{100}+0.07\)
asked 2021-02-11
Write an equation in standard form \(Ax=By=C\) of the line that satisfies the given conditions. Use integer values for A, B, and C. passes through (2, 8) and (-4, 16)
asked 2020-11-03
What is the solution to the following system of equations?
3x-6y=-12
x-2y=-8
(a)Use the substitution method to justify that the given system of equations has no solution.
(b)What do you know about the two lines in this system of equations?
...