Question
If w is the vector projection o [2,-3,1] onto [1,2,5], find the components of w.
Answers
GPT 3.5
We can find the components of w by using the formula for vector projection, which is given by:
proj_v(u) = (u · v / ||v||^2) * v
where u and v are vectors, · denotes the dot product, and ||v|| represents the magnitude of v.
Given:
u = [2, -3, 1]
v = [1, 2, 5]
First, find the dot product of u and v:
u · v = (2*1) + (-3*2) + (1*5) = 2 - 6 + 5 = 1
Next, find the magnitude squared of v:
||v||^2 = (1)^2 + (2)^2 + (5)^2 = 1 + 4 + 25 = 30
Now, plug these values into the formula for vector projection:
proj_v(u) = (1 / 30) * [1, 2, 5]
Therefore, the components of w are [1/30, 2/30, 5/30] which simplifies to [1/30, 1/15, 1/6].
proj_v(u) = (u · v / ||v||^2) * v
where u and v are vectors, · denotes the dot product, and ||v|| represents the magnitude of v.
Given:
u = [2, -3, 1]
v = [1, 2, 5]
First, find the dot product of u and v:
u · v = (2*1) + (-3*2) + (1*5) = 2 - 6 + 5 = 1
Next, find the magnitude squared of v:
||v||^2 = (1)^2 + (2)^2 + (5)^2 = 1 + 4 + 25 = 30
Now, plug these values into the formula for vector projection:
proj_v(u) = (1 / 30) * [1, 2, 5]
Therefore, the components of w are [1/30, 2/30, 5/30] which simplifies to [1/30, 1/15, 1/6].