Question

Vectors V_1 and V_2 are different vectors with lengths V1 and V2 respectively.

Vectors
ANSWERED
asked 2021-06-04

Vectors \(V_1\) and \(V_2\) are different vectors with lengths \(V_1\) and \(V_2\) respectively. Find the following:
a) \(V_1\cdot V_1\) Express you answer in terms of \(V_1\)
b) \(V_1\cdot V_2\), when they are perpendicular
c) \(V_1\cdot V_2\), when they are parallel

Answers (1)

2021-06-05

Dot product of two vectors is equal to the magnitude of first vector multiplied by magnitude of second vector multiplied by the cosine angle between the two vectors.
\(\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta\)
a) \(\vec{V_1}\cdot\vec{V_1}=|\vec{V_1}|\cdot|\vec{V_1}|\cos0^\circ\)
\(=(V_1)(V_1)\)
\(=V_1^2\)
b) \(\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos\theta\)
If the two vectors are perpendicular to each other, then \(\theta=90^\circ\)
Therefore,
\(\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos90^\circ\)
\(=(V_1)(V_2)(0)\)
\(=0\)
d) \(\vec{V_1}\cdot\vec{V_2}=|\vec{V_1}|\cdot|\vec{V_2}|\cos\theta\)
If the two vectors are parallel, the \(\theta=0^\circ\)
Therefore
\(\vec{V_1}\cdot|\vec{V_1}|\cdot|\vec{V_2}|\cos0^\circ\)
\(=(V_1)(V_2)(1)\)
\(=V_1V_2\)

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