Find the angle between $2 i + 5 j$ and $j$ .
The angle between $2 i + 5 j$ and $j$ is (Round to the nearest tenth as needed.)

Answer

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Answer

Final Answer: The angle between $2 i + 5 j$ and $j$ is $21.8$ degrees.

Steps

Step 1 ：Given vectors $A = 2 i + 5 j$ and $B = j$ .

Step 2 ：The dot product of $A$ and $B$ is calculated as $A \cdot B = 2 * 0 + 5 * 1 = 5$ .

Step 3 ：The magnitude of $A$ is calculated as $| | A | | = \sqrt{2^{2} + 5^{2}} = \sqrt{29}$ .

Step 4 ：The magnitude of $B$ is calculated as $| | B | | = \sqrt{0^{2} + 1^{2}} = 1$ .

Step 5 ：Using the dot product formula, we find $\cos (θ) = \frac{5}{\sqrt{29} * 1} = \frac{5}{\sqrt{29}}$ .

Step 6 ：Finally, we find the angle $θ$ by taking the inverse cosine of $\cos (θ)$ , which gives us $θ = 21.8$ degrees.

Step 7 ：Final Answer: The angle between $2 i + 5 j$ and $j$ is $21.8$ degrees.