The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

For what values of \(b\) are the two vectors \( (-4, b) \) and \( (b, {b}^{2} ) \) orthogonal?

Which of the following are correct where \(\vec{a}\) and \(\vec{b}\) are arbitrary vectors?

**I.** \(\lvert\vec{a}\cdot\vec{b}\rvert\le\lvert\vec{a}\rvert\lvert\vec{b}\rvert\)

**II.** \(\lvert\vec{a}+\vec{b}\rvert\le\lvert\vec{a}\rvert+\lvert\vec{b}\rvert\)

**III.** \(\lvert\vec{a}-\vec{b}\rvert\le\lvert\vec{a}\rvert-\lvert\vec{b}\rvert\)

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