QQuestionMathematics
QuestionMathematics
What is the angle between the vectors
$\vec{v}=\langle 2,5,7\rangle$ and
$\vec{w}=\langle 3,2,- 3\rangle$ ?
The angle is between 90 and 180 degrees.
They are perpendicular ( 90 degrees).
The vectors are parallel (zero angle between them).
The angle is between 0 and 90 degrees.
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Answer
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Step 1:: Calculate the dot product of the vectors v and w.
The dot product of two vectors, denoted as v · w, is given by the formula: v \cdot w = ||v|| ||w|| cos(\theta) where ||v|| and ||w|| are the magnitudes (or lengths) of vectors v and w, respectively, and θ is the angle between them. So, first, we need to find the magnitudes of v and w.
Step 2:: Find the magnitudes of the vectors v and w.
||\vec{v}|| = \sqrt{2^2 + 5^2 + 7^2} = \sqrt{4 + 25 + 49} = \sqrt{78} ||\vec{w}|| = \sqrt{3^2 + 2^2 + (- 3)^2} = \sqrt{9 + 4 + 9} = \sqrt{22}
Step 3:: Calculate the dot product of v and w.
\vec{v} \cdot \vec{w} = (2)(3) + (5)(2) + (7)(- 3) = 6 + 10 - 21 = - 5
Step 4:: Solve for cos(θ).
Recall the formula from Step 1: v \cdot w = ||v|| ||w|| cos(\theta) We can rearrange this formula to solve for cos(θ): Now, substitute the values we calculated in Steps 2 and 3:
Step 5:: Find the angle θ.
Now that we have the value of cos(θ), we can find the angle θ using the inverse cosine function (also called arccos). However, since we know that the angle is between 90 and 180 degrees, we should take the absolute value of cos(θ) before finding the inverse cosine:
Final Answer
The angle between the vectors v and w is approximately 104.53 degrees, which is between 90 and 180 degrees.
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