determine vectors u and v are parallel ,orthogonal or neither
u=(2,5) v=(10/3,4/3)
please use ab=|a|•|b|•cos theta to solve this problem!!! thx
c'mon. They are orthogonal if u•v = 0
And we know you can do dot products.
parallel if u•v = |u| * |v|
Come back with your work if you get stick.
but we don't know cos theta. How to calculate dot product?
Ummmmhhhh,
eg.
(a,b)•(c,d) = ac + bd
you have the numbers.
btw, your question says:
ab=|a|•|b|•cos
bad notation used here
it should be
a•b = |a||b|cos theta
i and j are x and y directions
if u = ux i + uy j
and v = vx i + vy j
then
u dot v = ux vx + uy vy
like the second way here
https://www.mathsisfun.com/algebra/vectors-dot-product.html
To determine whether vectors u and v are parallel, orthogonal, or neither, we can use the dot product formula:
u · v = |u| |v| cosθ
Here, u · v represents the dot product of vectors u and v, |u| and |v| represent the magnitudes (lengths) of vectors u and v respectively, and θ represents the angle between the vectors.
Step 1: Calculate the magnitudes:
|u| = √(2² + 5²) = √(4 + 25) = √29
|v| = √((10/3)² + (4/3)²) = √(100/9 + 16/9) = √((100+16)/9) = √116/3 = √(4 × 29/3) = (2/√3)√29
Step 2: Calculate the dot product:
u · v = (2 × 10/3) + (5 × 4/3) = (20/3 + 20/3) = 40/3
Step 3: Calculate the angle:
u · v = |u| |v| cosθ
40/3 = √29 × (2/√3)√29 × cosθ
40/3 = (2/√3)(29) cosθ
40/3 = (58/√3) cosθ
40/3 = (58/√3) cosθ
9.63 = 58 cosθ
cosθ = 9.63/58
cosθ = 0.16638
Step 4: Determine the relationship between the vectors based on the angle:
Since cosθ = 0.16638 is a positive value, the angle θ is between 0 and 90 degrees.
If cosθ = 0, the vectors u and v are orthogonal (perpendicular) to each other.
If cosθ < 0, the vectors u and v are neither parallel nor orthogonal.
If cosθ > 0, the vectors u and v are parallel to each other.
In this case, since cosθ = 0.16638 is positive, the vectors u and v are parallel to each other.