Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 05, 2023

Find the dot product of the vectors $\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}$ and $\mathbf{v} = 5\mathbf{j}$ .

STEP 1

Assumptions
1. The vectors are in two-dimensional space.
2. The vector $\mathbf{u}$ is represented as $2\mathbf{i} - 3\mathbf{j}$ .
3. The vector $\mathbf{v}$ is represented as $5\mathbf{j}$ .
4. The dot product of two vectors $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}$ is given by $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$ .

STEP 2

Write down the components of each vector.
For $\mathbf{u}$ , the components are: $u_1 = 2, u_2 = -3$
For $\mathbf{v}$ , the components are: $v_1 = 0, v_2 = 5$

STEP 3

Use the formula for the dot product to calculate $\mathbf{u} \cdot \mathbf{v}$ .
$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$

STEP 4

Substitute the components of $\mathbf{u}$ and $\mathbf{v}$ into the dot product formula.
$\mathbf{u} \cdot \mathbf{v} = (2)(0) + (-3)(5)$

STEP 5

Perform the multiplication for each term.
$\mathbf{u} \cdot \mathbf{v} = 0 + (-15)$

STEP 6

Add the terms to find the dot product.
$\mathbf{u} \cdot \mathbf{v} = -15$
The dot product $\mathbf{u} \cdot \mathbf{v}$ is $-15$ .

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