Let L be the line that passes through the points (0,1,6) and (0,3,2). Find the length of the projection of k=<0,0,1> on the line L.
I know how to do this if one part of the vector k is touching the line, but unless i'm mistaken, that is not the case in this question. Your help is very much appreciated!
To find the length of the projection of k = <0,0,1> on the line L that passes through the points (0,1,6) and (0,3,2), we can follow these steps:
Step 1: Find the direction vector of the line L
To find the direction vector of the line L, we can subtract the coordinates of the two given points. Let's call the direction vector D = <a, b, c>.
D = (0,3,2) - (0,1,6)
= <0-0, 3-1, 2-6>
= <0, 2, -4>
So, the direction vector of the line L is D = <0, 2, -4>.
Step 2: Find the unit vector in the direction of the line L
To find the unit vector in the direction of the line L, we need to normalize the direction vector D. We divide each component of D by its magnitude, which can be calculated using the formula:
|D| = sqrt(a^2 + b^2 + c^2)
|D| = sqrt(0^2 + 2^2 + (-4)^2)
= sqrt(0 + 4 + 16)
= sqrt(20)
= 2sqrt(5)
Now, we can find the unit vector u in the direction of D by dividing each component of D by |D|:
u = D / |D|
= <0 / (2sqrt(5)), 2 / (2sqrt(5)), -4 / (2sqrt(5))>
= <0, sqrt(5)/5, -2sqrt(5)/5>
So, the unit vector in the direction of the line L is u = <0, sqrt(5)/5, -2sqrt(5)/5>.
Step 3: Find the projection of k on the line L
The projection of k onto the line L is given by the dot product of k and u, multiplied by u:
projection = dot(k, u) * u
= (0 * 0) + (0 * sqrt(5)/5) + (1 * -2sqrt(5)/5)
= -2sqrt(5)/5
Therefore, the length of the projection of k = <0,0,1> on the line L is -2sqrt(5)/5.