Question

(a) find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and (b) find the vector component of u orthogonal to v. $$ \mathbf{u}=\langle 2,-3\rangle, \quad \mathbf{v}=\langle 3,2\rangle $$

Short answer

The projection of u onto v is \( \langle0,0\rangle \) and the vector component of u orthogonal to v is \( \langle2,-3\rangle \).

Step-by-step solution

Calculate the Dot Product of u and v

The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by \( \mathbf{u} \cdot \mathbf{v} = \) sum of the product of their corresponding components. In this case, \( \mathbf{u} \cdot \mathbf{v} = (2*3) + (-3*2) = 0.

Find the Square of V's Magnitude

We need to find the square of magnitude of vector \( \mathbf{v} \). The magnitude of a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \) is given by \( \| \mathbf{v} \|^2 = v_1^2 + v_2^2 \). Here, \( \| \mathbf{v} \|^2 = (3*3) + (2*2) = 13.

Calculate the Projection of u onto v

The projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is given by \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{ \mathbf{u} \cdot \mathbf{v}}{\| \mathbf{v} \|^2} * \mathbf{v} \). Substituting values we have \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{0}{13} * \langle3,2\rangle = \langle0,0\rangle \).

Find the Vector Component of u Orthogonal to v

The vector component of \( \mathbf{u} \) orthogonal to \( \mathbf{v} \) is given by \( \mathbf{u} - \text{proj}_{\mathbf{v}} \mathbf{u} \). Substituting values get \( \langle2,-3\rangle - \langle0,0\rangle = \langle2,-3\rangle \).