What is the angle between the given vector and the positive direction of the $x$-axis? (Round your answer to the nearest degree.) $\mathbf{i} + \sqrt{13} \mathbf{j}$.

Neetesh Kumar | December 18, 2024

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This is the solution to Math 1C
Assignment: 12.2 Question Number 9
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Step-by-step solution:

The angle $\theta$ between a vector $\mathbf{v} = \langle v_1, v_2 \rangle$ and the positive $x$-axis is given by:
$\theta = \arctan\left(\frac{v_2}{v_1}\right)$,
where $v_1$ is the $x$-component and $v_2$ is the $y$-component of the vector.

Step 1: Identify the components of $\mathbf{v}$:

The vector $\mathbf{v} = \mathbf{i} + \sqrt{13} \mathbf{j}$ can be expressed as:
$\mathbf{v} = \langle 1, \sqrt{13} \rangle.$

Here:

• $v_1 = 1$ ($x$-component),
• $v_2 = \sqrt{13}$ ($y$-component).

Step 2: Use the formula to find the angle:

Substitute $v_1 = 1$ and $v_2 = \sqrt{13}$ into the formula:
$\theta = \arctan\left(\frac{\sqrt{13}}{1}\right).$

Simplify:
$\theta = \arctan(\sqrt{13}).$

Step 3: Compute the angle:

Using a calculator, evaluate $\arctan(\sqrt{13})$:
$\theta \approx 73.90^\circ.$

Round to the nearest degree:
$\theta \approx 74^\circ.$

Final Answer:

The angle between the given vector and the positive direction of the $x$-axis is:

$\theta = \boxed{74^\circ}$

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