Resolve vector u =[3,4,7] into rectangular components, on which is collinear with vector v =[1,2,3].
DSA
To resolve vector u=[3,4,7] into rectangular components, one of which is collinear with vector v=[1,2,3], we can use the concept of projection.
1. Calculate the dot product of u and v:
u ⋅ v = (3 * 1) + (4 * 2) + (7 * 3) = 3 + 8 + 21 = 32
2. Calculate the magnitude of vector v:
|v| = √(1^2 + 2^2 + 3^2) = √(1 + 4 + 9) = √14 ≈ 3.74
3. Calculate the projection of u onto v:
proj_u_v = (u ⋅ v) / |v| = 32 / 3.74 ≈ 8.56
4. Calculate the unit vector in the direction of v:
u_v = v / |v| = [1/3.74, 2/3.74, 3/3.74] ≈ [0.27, 0.53, 0.80]
5. Calculate the rectangular component of u collinear with v:
component_u_v = proj_u_v * u_v = [0.27 * 8.56, 0.53 * 8.56, 0.80 * 8.56] ≈ [2.32, 4.54, 6.85]
Therefore, the rectangular component of vector u=[3,4,7] collinear with vector v=[1,2,3] is approximately [2.32, 4.54, 6.85].
To resolve vector u =[3,4,7] into rectangular components, we need to find the components of vector u that are collinear with vector v =[1,2,3].
The rectangular components of a vector are the projections of the vector onto the coordinate axes. To find the component of u that is collinear with v, we need to project u onto the direction of v.
The projection of vector u onto vector v can be calculated using the formula:
proj_u_v = (u . v) / ||v||
where u . v represents the dot product of u and v, and ||v|| represents the magnitude (or length) of v.
Let's calculate it step by step:
Step 1: Calculate the dot product of u and v:
u . v = (3 * 1) + (4 * 2) + (7 * 3) = 3 + 8 + 21 = 32
Step 2: Calculate the magnitude of v:
||v|| = sqrt((1^2) + (2^2) + (3^2)) = sqrt(1 + 4 + 9) = sqrt(14)
Step 3: Calculate the projection of u onto v:
proj_u_v = (u . v) / ||v|| = 32 / sqrt(14)
Now, the rectangular components of u that are collinear with v can be found by multiplying the projection of u onto v by the unit vector in the direction of v.
Let's find the unit vector in the direction of v:
unit_v = v / ||v|| = [1/sqrt(14), 2/sqrt(14), 3/sqrt(14)]
Finally, multiply the projection of u onto v by the unit vector in the direction of v to find the rectangular components of u that are collinear with v:
rectangular_components = proj_u_v * unit_v
Hence, you can compute the rectangular components using the above information.