Consider a vector a=[a1,a2]. Let's say that we compute the dot product of the vector with itself, i.e., compute a.a. What is the relationship between a.a and the magnitude of the vector ||a||?a.a = ||a||a.a = ||a||2a.a = √||a||
Question
Consider a vector a=[a1,a2]. Let's say that we compute the dot product of the vector with itself, i.e., compute a.a. What is the relationship between a.a and the magnitude of the vector ||a||?a.a = ||a||a.a = ||a||2a.a = √||a||
Solution
The dot product of a vector with itself, denoted as a.a, is equal to the square of the magnitude of the vector. In mathematical terms, this is expressed as a.a = ||a||^2.
Here's the step-by-step explanation:
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The dot product of a vector with itself is calculated as a1a1 + a2a2.
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The magnitude of a vector is calculated as the square root of the sum of the squares of its components, i.e., ||a|| = sqrt(a1^2 + a2^2).
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Squaring the magnitude of the vector gives us a1^2 + a2^2, which is the same as the dot product of the vector with itself.
So, a.a = ||a||^2, not ||a|| or √||a||.
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