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

\(\vec{V_1}\cdot|\vec{V_1}|\cdot|\vec{V_2}|\cos0^\circ\)

\(=(V_1)(V_2)(1)\)

\(=V_1V_2\)