Determine the projection of u = 1.6i + 3.3j in the v = -2.1i - 0.5j direction
a. 0.8i +0.2j
b. 2.3i = 0.5j
c.-0.6i-1.3j
d. -1.2i-1.1j
To find the angle between u and v,
|u|*|v| cosθ = u•v
3.67*2.16 cosθ = (1.6)(-2.1)+(3.3)(-0.5)
cosθ = -5.01/7.93 = -0.63
θ = 129.18°
So the projection of u on v is
|u|cosθ v/|v|
= 3.67(-0.63)/2.16 v
= -1.07v
≈ 2.3i+0.5j
or, without finding the angle,
projection of u on v
= (u dot v)/|v|^2 * v
= (1.6*-2.1 + 3.3*-.5)/√(4.41+.25)^2 * v
= appr -1.0796 v
= appr 2.26i + .539j ----> given as choice b), same as Steve's answer
To determine the projection of vector u onto vector v, we will use the formula:
proj_v(u) = (u · v̂)v̂
where u · v̂ is the dot product between u and the unit vector v̂.
First, let's find the unit vector v̂:
||v|| = √((-2.1)^2 + (-0.5)^2)
= √(4.41 + 0.25)
= √4.66
≈ 2.16
v̂ = ( -2.1/2.16, -0.5/2.16 )
≈ (-0.972, -0.231)
Now, calculate the dot product u · v̂:
u · v̂ = (1.6 * -0.972) + (3.3 * -0.231)
≈ -1.555 + (-0.762)
≈ -2.317
Finally, multiply u · v̂ with v̂ to find the projection:
proj_v(u) ≈ -2.317 * (-0.972, -0.231)
≈ (2.253, 0.534)
Therefore, the projection of u onto v is approximately 2.253i + 0.534j.
The closest option is:
b. 2.3i + 0.5j
To determine the projection of vector u onto vector v, we can use the formula for the projection:
projv(u) = ((u · v) / ||v||^2) * v
Where:
- u · v denotes the dot product of u and v
- ||v|| denotes the magnitude of vector v
First, let's calculate the dot product of u and v:
u · v = (1.6)(-2.1) + (3.3)(-0.5)
= -3.36 - 1.65
= -5.01
Next, let's calculate the magnitude of vector v:
||v|| = sqrt((-2.1)^2 + (-0.5)^2)
= sqrt(4.41 + 0.25)
= sqrt(4.66)
≈ 2.16
Now, we can plug these values into the projection formula:
projv(u) = ((u · v) / ||v||^2) * v
= (-5.01 / (2.16)^2) * (-2.1i - 0.5j)
To find the result, simplify the expression:
projv(u) ≈ 0.796i + 0.199j
So, the projection of vector u onto the v direction is approximately 0.796i + 0.199j.
Comparing this to the answer choices, we can see that the option closest to the projection we calculated is:
a. 0.8i + 0.2j
Therefore, the correct option is a. 0.8i + 0.2j.