# 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 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 $$v = ⟨4, 7, 0⟩$$

2020-12-08
Step 1 Given: The line through (0, 0, 1) in the direction of the vector$$v = <4, 7, 0>$$. To find: Parametric and vector equations of the given line. Step 2 Let, $$\overrightarrow{r_0} = (0, 0, 1), \overrightarrow{d} = <4, 7, 0>$$ The vector equation of the line is, $$\overrightarrow{r} = \overrightarrow{r_0} + t \overrightarrow{d}$$
$$\Rightarrow \overrightarrow{r} = <0, 0, 1> + t <4, 7, 0>$$
$$= <0, 0, 1> + <4t, 7t, 0>$$
$$= <0 + 4t, 0 + 7t, 1 + 0>$$
$$= <4t, 7t, 1>$$ $$\Rightarrow \overrightarrow{r} = <4t, 7t, 0>$$ The parametric equations of a line are, Result :$$x = 4t, y = 7t, z = 0$$ Vector equation of a line is, $$\overrightarrow{r} = <4t, 7t, 0>$$ Parametric equations of a line are, $$x = 4t, y = 7t, z = 0.$$

### Relevant Questions

Find a unit vector that is orthogonal to both $$i+j$$ and $$i+k$$.
Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:
$$f(x,y)=\begin{cases}xe^{-x(1+y)} & x\geq0\ and\ \geq0\\ 0 & otherwise \end{cases}$$
a) What is the probability that the lifetime X of the first component exceeds 3?
b) What are the marginal pdf's of X and Y? Are the two lifetimes independent? Explain.
c) What is the probability that the lifetime of at least one component exceeds 3?
Evaluate the following integral in cylindrical coordinates triple integral $$\int_{x=-1}^{1}\int_{y=0}^{\sqrt{1-x^{2}}}\int_{z=0}^{2}\left(\frac{1}{1+x^{2}+y^{2}}\right)dzdxdy$$
Find an equation of the plane tangent to the following surface at the given point.
$$7xy+yz+4xz-48=0;\ (2,2,2)$$
Find an equation of the plane.
The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)
Find the length of the curve.
$$r(t)=\langle6t,\ t^{2},\ \frac{1}{9}t^{3}\rangle,\ 0\leq t\leq1$$
Use the table of values of $$f(x, y)$$ to estimate the values of $$fx(3, 2)$$, $$fx(3, 2.2)$$, and $$fxy(3, 2)$$.
$$\begin{array}{|c|c|}\hline y & 1.8 & 2.0 & 2.2 \\ \hline x & & & \\ \hline 2.5 & 12.5 & 10.2 & 9.3 \\ \hline 3.0 & 18.1 & 17.5 & 15.9 \\ \hline 3.5 & 20.0 & 22.4 & 26.1 \\ \hline \end{array}$$
Find the points on the ellipse $$4x^2 + y^2 = 4$$ that are farthest away from the point (-1, 0).
$$u=(-3,\ 9,\ 6)$$
$$v=(4,\ -12,\ -8)$$
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