Determine if the following vectors are orthogonal.

$\mathbf{u} = \begin{bmatrix} 7 \\ 2 \\ -3 \\ 0 \end{bmatrix}, \mathbf{v} = \begin{bmatrix} -4 \\ 11 \\ -2 \\ 5 \end{bmatrix}$

Are the two vectors orthogonal?
(Type an integer or a fraction.)

Neetesh Kumar | October 18, 2024

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This is the solution to Math2B Course: Linear Algebra
Assignment: Ch6 Section 1 Question Number 3
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Step-by-step solution:

Two vectors are orthogonal if their dot product is zero.
The dot product of two vectors $\mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix}$ is given by:

$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 + u_4 v_4$

Step 1: Calculate the dot product

For the given vectors $\mathbf{u} = \begin{bmatrix} 7 \\ 2 \\ -3 \\ 0 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} -4 \\ 11 \\ -2 \\ 5 \end{bmatrix}$, the dot product is:

$\mathbf{u} \cdot \mathbf{v} = (7)(-4) + (2)(11) + (-3)(-2) + (0)(5)$

Simplify each term:

$(7)(-4) = -28$
$(2)(11) = 22$
$(-3)(-2) = 6$
$(0)(5) = 0$

Now, sum up the results:

$\mathbf{u} \cdot \mathbf{v} = -28 + 22 + 6 + 0 = 0$

Step 2: Determine if the vectors are orthogonal

Since the dot product is 0, the vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.

Final Answer

The vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal because $\mathbf{u} \cdot \mathbf{v} = 0$.

• Option - D. The vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal because $\mathbf{u} \cdot \mathbf{v} = \boxed{0}$.

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