the scalar equation of the line through (-1,5) if (4,-1) is a normal vector to the line.
To find the scalar equation of a line through a given point with a normal vector, we can use the formula:
Ax + By = C
Where A, B, and C are constants, and (x, y) are the coordinates of any point on the line.
Given that the point (-1, 5) lies on the line and the vector (4, -1) is a normal vector to the line, we can determine the equation of the line.
Step 1: Finding the values of A, B, and C
To begin, we substitute the coordinates (-1, 5) into the equation:
A(-1) + B(5) = C
Simplifying:
-A + 5B = C
Step 2: Finding A and B using the normal vector
Since (4, -1) is a normal vector to the line, we know that any line perpendicular to this vector will have a direction vector parallel to (4, -1). Therefore, the direction vector of the line is also (4, -1).
Using this information, we can substitute the values of A and B into the equation for our normal vector:
A = 4
B = -1
Step 3: Generating the scalar equation
Now that we have the values for A and B, we substitute them back into the general equation:
-4 + 5(-1) = C
Simplifying:
-4 - 5 = C
C = -9
Therefore, the scalar equation of the line passing through (-1, 5) with the normal vector (4, -1) is:
4x - y = -9