find the scalar projection of u onto v, vector projection and the vector component of u orthogonal to v
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The Correct Answer and Explanation is :
To find the scalar projection of vector u onto vector v, the vector projection of u onto v, and the vector component of u orthogonal to v, we need to follow certain steps. Let’s define u and v as vectors.
Let:
- u = (u₁, u₂, u₃) (a 3D vector),
- v = (v₁, v₂, v₃) (another 3D vector).
1. Scalar Projection of u onto v
The scalar projection of u onto v (also called the “component” of u in the direction of v) is given by the formula:
[
\text{Scalar projection} = \text{comp}_v(u) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}
]
Where:
- u · v is the dot product of u and v:
[
\mathbf{u} \cdot \mathbf{v} = u₁v₁ + u₂v₂ + u₃v₃
] - ||v|| is the magnitude of v:
[
|\mathbf{v}| = \sqrt{v₁^2 + v₂^2 + v₃^2}
]
2. Vector Projection of u onto v
The vector projection of u onto v is given by:
[
\text{Vector projection} = \mathbf{proj}_v(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \mathbf{v}
]
This represents the vector along v in the same direction as u.
3. Vector Component of u Orthogonal to v
The component of u orthogonal to v is found by subtracting the vector projection of u onto v from u:
[
\mathbf{u}_{\text{orth}} = \mathbf{u} – \mathbf{proj}_v(\mathbf{u})
]
This is the part of u that is perpendicular to v.
Explanation
- The scalar projection tells us how much of vector u lies in the direction of v (it’s a scalar value).
- The vector projection represents the part of u that is aligned with v.
- The orthogonal component is the part of u that is perpendicular to v, and it’s what’s left when you subtract the projection from u.
These projections and components are important in various applications, such as in physics for resolving forces, in computer graphics for rendering shadows, and in geometry for understanding vector relationships.
I’ll now generate an image to visually demonstrate these concepts. Please provide the specific vectors u and v so I can calculate the projections.