Questions: For u=⟨4,-5⟩ and v=⟨-3,5⟩ find u · v.

Transcript text: For $\mathbf{u}=\langle 4,-5\rangle$ and $\mathbf{v}=\langle-3,5\rangle$ find $\mathbf{u} \cdot \mathbf{v}$.

Solution

Solution Steps

To find the dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$, we multiply their corresponding components and then sum the results. Specifically, for vectors $\mathbf{u} = \langle u_1, u_2 \rangle$ and $\mathbf{v} = \langle v_1, v_2 \rangle$, the dot product $\mathbf{u} \cdot \mathbf{v}$ is calculated as $u_1 \cdot v_1 + u_2 \cdot v_2$.

Step 1: Define the Vectors

We are given the vectors $\mathbf{u} = \langle 4, -5 \rangle$ and $\mathbf{v} = \langle -3, 5 \rangle$.

Step 2: Calculate the Dot Product

The dot product $\mathbf{u} \cdot \mathbf{v}$ is calculated using the formula: \[ \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \] Substituting the values: \[ \mathbf{u} \cdot \mathbf{v} = 4 \cdot (-3) + (-5) \cdot 5 \]

Step 3: Perform the Multiplications

Calculating each term: \[ 4 \cdot (-3) = -12 \] \[ (-5) \cdot 5 = -25 \]

Step 4: Sum the Results

Now, we sum the results of the multiplications: \[ \mathbf{u} \cdot \mathbf{v} = -12 + (-25) = -37 \]