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MTLE Physics (064, 065) Practice Tests & Test Prep by Exam Edge


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MTLE 064, 065 Practice Test Features

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MTLE Physics (9-12) Sample Test

1 of 5

What is equal to the dot product of a vector with itself?





Correct Answer:
the magnitude of the vector squared


the dot product, also known as the scalar product, of a vector with itself has a specific mathematical outcome. when calculating the dot product of any vector \(\vec{a}\) with itself, we use the formula \(\vec{a} \cdot \vec{a}\). this formula can be expanded based on the definition of the dot product: \(\vec{a} \cdot \vec{a} = ||\vec{a}|| \times ||\vec{a}|| \times \cos(\theta)\), where \(\theta\) is the angle between the two vectors and \(||\vec{a}||\) represents the magnitude (or length) of the vector \(\vec{a}\).

in the case where the vector is dotted with itself, the angle \(\theta\) is 0 degrees because both vector directions are identical. the cosine of 0 degrees is 1. hence, the formula simplifies to \(\vec{a} \cdot \vec{a} = ||\vec{a}|| \times ||\vec{a}|| \times 1 = ||\vec{a}||^2\). this result indicates that the dot product of a vector with itself equals the square of its magnitude.

therefore, the correct response to what the dot product of a vector with itself equals is 'the magnitude of the vector squared'. this outcome is a fundamental property in vector mathematics, pivotal in various applications including physics and engineering, where it often relates to calculations involving power, energy, or projections.
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