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

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

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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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