Questions: Find a · b. a = ⟨4, 1, 1/5⟩, b = ⟨8, -5, -15⟩ Find a · b. a = 7i + j, b = i - 6j + k

Transcript text: Find $\mathbf{a} \cdot \mathbf{b}$. \[ \mathbf{a}=\left\langle 4,1, \frac{1}{5}\right\rangle, \quad \mathbf{b}=\langle 8,-5,-15\rangle \] Find $\mathbf{a} \cdot \mathbf{b}$. \[ \mathbf{a}=7 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{i}-6 \mathbf{j}+\mathbf{k} \]

Solution

Solution Steps

To find the dot product of two vectors, we multiply corresponding components of the vectors and then sum the results.

First Question:

Given vectors: \[ \mathbf{a}=\left\langle 4,1, \frac{1}{5}\right\rangle, \quad \mathbf{b}=\langle 8,-5,-15\rangle \] The dot product $\mathbf{a} \cdot \mathbf{b}$ is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = 4 \cdot 8 + 1 \cdot (-5) + \frac{1}{5} \cdot (-15) \]

Step 1: Understanding the Problem

We are given two vectors $\mathbf{a}$ and $\mathbf{b}$ and need to find their dot product $\mathbf{a} \cdot \mathbf{b}$.

Step 2: Define the Vectors

The vectors are: \[ \mathbf{a} = \left\langle 4, 1, \frac{1}{5} \right\rangle, \quad \mathbf{b} = \langle 8, -5, -15 \rangle \]

Step 3: Dot Product Formula

The dot product of two vectors $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ is given by: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

Step 4: Substitute the Components

Substitute the components of $\mathbf{a}$ and $\mathbf{b}$ into the dot product formula: \[ \mathbf{a} \cdot \mathbf{b} = 4 \cdot 8 + 1 \cdot (-5) + \frac{1}{5} \cdot (-15) \]

Step 5: Perform the Multiplications

Calculate each term separately: \[ 4 \cdot 8 = 32 \] \[ 1 \cdot (-5) = -5 \] \[ \frac{1}{5} \cdot (-15) = -3 \]

Step 6: Sum the Results

Add the results of the multiplications: \[ 32 + (-5) + (-3) = 32 - 5 - 3 = 24 \]