Question

let \\( \mathbf{u} = -8\mathbf{i} + 5\mathbf{j} \\) and \\( \mathbf{v} = 6\mathbf{i} - \mathbf{j} \\). find \\( \\| \mathbf{u} + \mathbf{v} \\|^2 - \\| \mathbf{u} - \mathbf{v} \\|^2 \\).
\\( \\| \mathbf{u} + \mathbf{v} \\|^2 - \\| \mathbf{u} - \mathbf{v} \\|^2 = \square \\)

Explanation:

Step1: Recall vector norm identity

Recall that for vectors $\mathbf{u}$ and $\mathbf{v}$, $\|\mathbf{u}+\mathbf{v}\|^2 - \|\mathbf{u}-\mathbf{v}\|^2 = 4\mathbf{u} \cdot \mathbf{v}$, where $\cdot$ is the dot product.

Step2: Compute the dot product $\mathbf{u} \cdot \mathbf{v}$

Given $\mathbf{u} = -8\mathbf{i} + 5\mathbf{j}$ and $\mathbf{v} = 6\mathbf{i} - \mathbf{j}$, the dot product is:

$$ \mathbf{u} \cdot \mathbf{v} = (-8)(6) + (5)(-1) = -48 - 5 = -53 $$

Step3: Calculate the final value

Multiply the dot product by 4:

$$ 4\mathbf{u} \cdot \mathbf{v} = 4(-53) = -212 $$

Answer:

$-212$