Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (9, 2); perpendicular to v = 4i + 4j
A) $4 x + 4 y = 44$
B) $4 x - 4 y = 28$
C) $4 x + 4 y = 32$
D) $y - 2 = - \frac { 2 } { 5 } ( x - 4 )$

Question

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (9, 2); perpendicular to v = 4i + 4j 
A) $4 x + 4 y = 44$
B) $4 x - 4 y = 28$
C) $4 x + 4 y = 32$
D) $y - 2 = - \frac { 2 } { 5 } ( x - 4 )$

Quizplus · Accepted Answer

The vector $v = 4i + 4j$ has a direction vector of $<4, 4>$, so a line perpendicular to it will have a slope that is the negative reciprocal of the slope of $v$. Since the slope of $v$ is $4/4 = 1$, a line perpendicular to it will have a slope of $-1$. However, since we are dealing with vectors and their perpendicularity, we can directly use the coefficients of $i$ and $j$ to find the equation of the line perpendicular to $v$. The line will have coefficients that are negative reciprocals of each other, but since we are looking for an equation and not the slope, we can use the dot product property where the dot product of perpendicular vectors is 0. Thus, a line perpendicular to $4i + 4j$ can be represented as $4x + 4y = C$. To find $C$, we use the given point $P = (9, 2)$, leading to $4(9) + 4(2) = 36 + 8 = 44$. Therefore, the equation of the line is $4x + 4y = 44$.

Find an Equation for the Line That Passes Through the Given