The vector equation of the line is $\newline$ $\begin{array}{c} \mathbf{r}= \\ (4-\lambda) \mathbf{i}-(3+2 \lambda) \mathbf{j}+\lambda \mathbf{k}, \lambda \in \mathbb{R} . \end{array}$ $\newline$ Then the Cartesian equation of the line is $\newline$ $\frac{x-p}{q}=\frac{y-r}{s}=z \text {. }$ $\newline$ Find the value of $p, q, r$ and $s$ .

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Algebra 1

Write a quadratic function from its x-intercepts and another point

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Q. The vector equation of the line is

\newline

\begin{array}{c} \mathbf{r}= \\ (4-\lambda) \mathbf{i}-(3+2 \lambda) \mathbf{j}+\lambda \mathbf{k}, \lambda \in \mathbb{R} . \end{array}

\newline

Then the Cartesian equation of the line is

\newline

\frac{x-p}{q}=\frac{y-r}{s}=z \text {. }

\newline

Find the value of

p, q, r

and

s

Given Vector Equation: The vector equation of the line is given by $\mathbf{r} = (4 - \lambda)\mathbf{i} - (3 + 2\lambda)\mathbf{j} + \lambda\mathbf{k}$ . To find the Cartesian equation, we compare this with the general vector equation of a line $\mathbf{r} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ , where $(x, y, z)$ are the coordinates of any point on the line and $(a, b, c)$ are the direction ratios.
Determine Direction Ratios: From the given vector equation, the direction ratios are $a = -1$ , $b = -2$ , and $c = 1$ . These will be used as the denominators in the Cartesian equation of the line.
Form Cartesian Equation: The Cartesian equation of the line is $(x - p)/q = (y - r)/s = z$ , where $p$ , $q$ , $r$ , and $s$ are constants to be determined.
Find Constants: To find $p$ , $q$ , $r$ , and $s$ , we look at the coefficients of $\lambda$ in the vector equation. For the $x$ -component, we have $4 - \lambda$ , which means when $\lambda = 0$ , $x = 4$ . Therefore, $p = 4$ and $q$ $0$ (since the direction ratio $q$ $1$ ).
Write Cartesian Equation: For the $y$ -component, we have $-3 - 2\lambda$ , which means when $\lambda = 0$ , $y = -3$ . Therefore, $r = -3$ and $s = -2$ (since the direction ratio $b = -2$ ).
Write Cartesian Equation: For the y-component, we have $-3 - 2\lambda$ , which means when $\lambda = 0$ , $y = -3$ . Therefore, $r = -3$ and $s = -2$ (since the direction ratio $b = -2$ ).For the z-component, we have $\lambda$ , which means when $\lambda = 0$ , $z = 0$ . However, since the direction ratio $c = 1$ , we don't need to find a constant for $\lambda = 0$ $0$ in the Cartesian equation.
Write Cartesian Equation: For the y-component, we have $-3 - 2\lambda$ , which means when $\lambda = 0$ , $y = -3$ . Therefore, $r = -3$ and $s = -2$ (since the direction ratio $b = -2$ ).For the z-component, we have $\lambda$ , which means when $\lambda = 0$ , $z = 0$ . However, since the direction ratio $c = 1$ , we don't need to find a constant for $\lambda = 0$ $0$ in the Cartesian equation.Now we can write the Cartesian equation of the line as $\lambda = 0$ $1$ .