resolve the vector v=[7, 9] into rectangular components, one of which is in the direction of [6, 2]
To resolve a vector into rectangular components, we need to find the components of the vector in the x and y directions.
Let's call the rectangular components of the vector v as vx and vy, and the direction vector as d.
Given:
v = [7, 9]
d = [6, 2]
First, we need to determine the magnitude of the direction vector d. We can use the Pythagorean theorem to find this:
|d| = sqrt(d[0]^2 + d[1]^2)
= sqrt(6^2 + 2^2)
= sqrt(36 + 4)
= sqrt(40)
= 2 * sqrt(10)
Next, we will normalize the direction vector d by dividing it by its magnitude:
normalized_d = d / |d|
= [6, 2] / (2 * sqrt(10))
= [6 / (2 * sqrt(10)), 2 / (2 * sqrt(10))]
= [3 / sqrt(10), 1 / sqrt(10)]
Now, we can find the x-component of the vector v in the direction of normalized_d:
vx = |v| * dot_product(v, normalized_d)
= sqrt(v[0]^2 + v[1]^2) * (v[0]*normalized_d[0] + v[1]*normalized_d[1])
= sqrt(7^2 + 9^2) * (7 * (3 / sqrt(10)) + 9 * (1 / sqrt(10)))
= sqrt(49 + 81) * (21 / sqrt(10) + 9 / sqrt(10))
= sqrt(130) * (30 / sqrt(10))
= sqrt(130) * (3 * sqrt(10) / (sqrt(10) * sqrt(10)))
= 3 * sqrt(130) * sqrt(10) / 10
= 3 * sqrt(1300) / 10
= 3 * 10 * sqrt(13) / 10
= 3 * sqrt(13)
Similarly, we can find the y-component of the vector v in the direction of normalized_d:
vy = |v| * dot_product(v, normalized_d)
= sqrt(7^2 + 9^2) * (7 * (1 / sqrt(10)) + 9 * (-3 / sqrt(10)))
= 3 * sqrt(130) * (-2 * sqrt(10)) / 10
= -3 * 2 * sqrt(1300) / 10
= -6 * sqrt(13)
Therefore, the rectangular components of the vector v, one of which is in the direction of d, are:
vx = 3 * sqrt(13)
vy = -6 * sqrt(13)
Hence, the vector v=[7, 9] resolved into rectangular components, one of which is in the direction of [6, 2], is [3 * sqrt(13), -6 * sqrt(13)].
To resolve a vector into rectangular components, we can use the concept of projections. To find the rectangular component of a vector in a particular direction, we need to find the projection of the vector onto that direction.
The projection of vector v onto direction d can be calculated using the following formula:
Projection of v onto d = (v dot d) / (magnitude of d)
Where "dot" denotes the dot product.
Now let's use this formula to resolve vector v=[7, 9] into rectangular components in the direction of [6, 2]:
1. Calculate the dot product of v and d:
v dot d = (7)(6) + (9)(2) = 42 + 18 = 60
2. Calculate the magnitude of d:
Magnitude of d = sqrt((6)^2 + (2)^2) = sqrt(36 + 4) = sqrt(40) = 2√10
3. Calculate the projection of v onto d:
Projection of v onto d = (v dot d) / (magnitude of d) = 60 / (2√10) = (60√10) / 20 = 3√10
Therefore, the rectangular component of vector v=[7, 9] in the direction of [6, 2] is 3√10.