If w is the vector [2,-3,1] onto [1,2,5], find the components of w
To find the components of vector w onto vector [1,2,5], we can use the formula:
proj_v(w) = ((w • v) / (v • v)) * v
where • represents the dot product of two vectors.
Step 1: Calculate the dot product (w • v):
w • v = 2*1 + (-3)*2 + 1*5 = 2 - 6 + 5 = 1
Step 2: Calculate the dot product (v • v):
v • v = 1*1 + 2*2 + 5*5 = 1 + 4 + 25 = 30
Step 3: Divide the dot product of w and v by the dot product of v and v:
(1/30) * [1,2,5] = [(1/30) * 1, (1/30) * 2, (1/30) * 5]
= [1/30, 1/15, 1/6]
So, the components of w onto [1,2,5] are [1/30, 1/15, 1/6].
To find the components of vector w onto vector v, we can use the projection formula.
The projection of w onto v is given by the formula:
projv(w) = (w . v) / ||v||^2 * v
where "." represents the dot product and "||v||" represents the magnitude of vector v.
Let's calculate the components of w onto v step-by-step.
1. Calculate the dot product between vectors w and v:
w . v = (2 * 1) + (-3 * 2) + (1 * 5) = 2 - 6 + 5 = 1.
2. Calculate the magnitude of vector v:
||v|| = sqrt((1^2) + (2^2) + (5^2)) = sqrt(1 + 4 + 25) = sqrt(30).
3. Calculate the projection of w onto v:
projv(w) = (w . v) / ||v||^2 * v = 1 / 30 * [1, 2, 5]
= [1/30, 2/30, 5/30]
= [1/30, 1/15, 1/6].
Therefore, the components of vector w onto vector v are [1/30, 1/15, 1/6].