QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
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}
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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