Resolve the vector v=[7,9] into rectangular components, one of which is in the direction of u=[6,2].
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To resolve vector v=[7,9] into rectangular components in the direction of vector u=[6,2], we need to find two components: one component parallel to vector u (in the same direction) and one component perpendicular to vector u.
To find the component parallel to vector u, we need to project vector v onto vector u. The formula for the parallel component P is given by:
P = (v dot u) / (u dot u) * u
where dot represents the dot product of two vectors.
Let's calculate the dot products first:
v dot u = 7*6 + 9*2 = 42 + 18 = 60
u dot u = 6*6 + 2*2 = 36 + 4 = 40
Now we can calculate the parallel component P:
P = (60 / 40) * [6, 2]
P = [1.5 * 6, 1.5 * 2]
P = [9, 3]
So, the component of vector v in the direction of vector u is [9, 3].
To find the component perpendicular to vector u, we can use the formula:
Perpendicular component S = v - P
S = [7, 9] - [9, 3]
S = [-2, 6]
Therefore, the component of vector v perpendicular to vector u is [-2, 6].
To summarize:
- The component of vector v in the direction of vector u is [9, 3].
- The component of vector v perpendicular to vector u is [-2, 6].
To resolve the vector v=[7,9] into rectangular components, one of which is in the direction of u=[6,2], we first need to find the projection of v onto u.
The projection of v onto u can be calculated using the formula: proj_u(v) = (v · u) / (||u||^2) * u
Here's how to obtain the result step by step:
Step 1: Calculate the dot product of v and u: (v · u)
(v · u) = (7 * 6) + (9 * 2)
= 42 + 18
= 60
Step 2: Calculate the magnitude squared of u: ||u||^2
||u||^2 = (6 * 6) + (2 * 2)
= 36 + 4
= 40
Step 3: Calculate the projection of v onto u: proj_u(v) = (v · u) / (||u||^2) * u
proj_u(v) = (60 / 40) * u
= 1.5 * [6, 2]
= [9, 3]
So, the resolved vector of v=[7,9] in the direction of u=[6,2] would be [9,3].