Find the component vectors.
v=(7i+j) onto w=(4i+j)
v_1=?
v_2=?
To find the component vectors of v onto w, we need to project v onto w. The projection of v onto w can be calculated using the formula:
proj_w(v) = (v • w) / ||w||^2 * w
Where:
- "•" denotes the dot product between v and w.
- "||w||^2" represents the magnitude of w squared.
- "w" is the unit vector in the direction of w.
Let's calculate the projection of v onto w.
Step 1: Calculate the dot product between v and w.
v • w = (7i + j) • (4i + j)
= 7*4(i • i) + 7*(i • j) + 1*(j • i) + 1*(j • j)
= 7*4(1) + 0 + 0 + 1(1)
= 28 + 1
= 29
Step 2: Calculate the magnitude of w squared.
||w||^2 = ||(4i + j)||^2
= (4i + j) • (4i + j)
= 4*4(i • i) + 4*(i • j) + 1*(j • i) + 1(1)
= 4*4(1) + 0 + 0 + 1(1)
= 16 + 1
= 17
Step 3: Calculate the projection of v onto w.
proj_w(v) = (v • w) / ||w||^2 * w
= (29 / 17) * (4i + j)
The component vectors of v onto w are given by multiplying the projection by the unit vector of w:
v_1 = (29 / 17) * 4i
v_2 = (29 / 17) * j
Therefore, the component vectors are:
v_1 = (29 / 17) * 4i
v_2 = (29 / 17) * j
To find the component vectors of v onto w, we need to determine how much of v is in the direction of w.
First, we need to find the scalar projection of v onto w. This can be calculated using the formula:
v_1 = (v · w) / |w|
where v_1 is the scalar projection of v onto w, v · w is the dot product of v and w, and |w| is the magnitude of w.
Let's calculate:
v · w = (7i + j) · (4i + j)
= 7*4 + 1*1
= 28 + 1
= 29
|w| = √(4^2 + 1^2)
= √(16 + 1)
= √17
Substituting these values into the formula:
v_1 = (29) / (√17)
= 29/√17
Next, we need to find the vector projection of v onto w. This can be calculated by multiplying the scalar projection (v_1) by the unit vector in the direction of w.
The unit vector in the direction of w can be found by dividing w by its magnitude:
u_w = w / |w|
Let's calculate:
u_w = (4i + j) / √17
Now, we can calculate the vector projection:
v_2 = v_1 * u_w
= (29/√17) * (4i + j)
= (29/√17) * 4i + (29/√17) * j
Therefore, the component vectors are:
v_1 = 29/√17
v_2 = (29/√17) * 4i + (29/√17) * j
if you mean the projection of v onto w, then that is just
v•w/|w|
So just crank it out. Post your work if you get stuck.