Find the projection of u onto v and then find the vector component of u orthogonal to v. Show the steps of your calculation. Sketch a figure and label u, v, the projection of u onto v and the vector component of u orthogonal to v. u=(2,3) and v=(5,1) proj, (u) = w = vector component of u orthogonal to V=
The Correct Answer and Explanation is :
To solve for the projection of vector u onto vector v, and the vector component of u orthogonal to v, we will follow these steps:
Step 1: Formula for the projection of u onto v
The formula for the projection of vector u onto vector v is given by:
[
\text{proj}_{v}(u) = \frac{u \cdot v}{v \cdot v} v
]
where:
- ( u \cdot v ) is the dot product of u and v
- ( v \cdot v ) is the dot product of v with itself, i.e., the squared magnitude of v
Step 2: Calculate the dot products
Given:
- u = (2, 3)
- v = (5, 1)
Dot product ( u \cdot v ):
[
u \cdot v = (2)(5) + (3)(1) = 10 + 3 = 13
]
Dot product ( v \cdot v ):
[
v \cdot v = (5)(5) + (1)(1) = 25 + 1 = 26
]
Step 3: Calculate the projection of u onto v
Using the formula:
[
\text{proj}_{v}(u) = \frac{13}{26} v = \frac{1}{2} (5, 1) = \left( \frac{5}{2}, \frac{1}{2} \right)
]
So, the projection of u onto v is ( \left( \frac{5}{2}, \frac{1}{2} \right) ).
Step 4: Calculate the component of u orthogonal to v
The component of u orthogonal to v can be found by subtracting the projection of u onto v from u:
[
w = u – \text{proj}_{v}(u)
]
[
w = (2, 3) – \left( \frac{5}{2}, \frac{1}{2} \right) = \left( 2 – \frac{5}{2}, 3 – \frac{1}{2} \right)
]
[
w = \left( \frac{4}{2} – \frac{5}{2}, \frac{6}{2} – \frac{1}{2} \right) = \left( -\frac{1}{2}, \frac{5}{2} \right)
]
So, the vector component of u orthogonal to v is ( \left( -\frac{1}{2}, \frac{5}{2} \right) ).
Step 5: Conclusion
- The projection of u onto v is ( \left( \frac{5}{2}, \frac{1}{2} \right) ).
- The vector component of u orthogonal to v is ( \left( -\frac{1}{2}, \frac{5}{2} \right) ).
Now, let’s create a sketch to visualize this:
I’ll generate a plot with the vectors u, v, the projection of u onto v, and the orthogonal component.
In the plot above, you can see the following:
- The red vector u = (2, 3).
- The blue vector v = (5, 1).
- The green vector represents the projection of u onto v (( \text{proj}_v(u) = \left( \frac{5}{2}, \frac{1}{2} \right) )).
- The purple vector represents the component of u orthogonal to v (( w = \left( -\frac{1}{2}, \frac{5}{2} \right) )).
This visualization helps to understand how u breaks down into two parts: one parallel to v (the projection) and one orthogonal to v (the orthogonal component).