Draw diagrams to show two vectors $\vec{a}$ and $\vec{b}$, and the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$.

Question

Draw diagrams to show two vectors $\mathbf{a}$ and $\mathbf{b}$, and the vectors $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$.

(a) When is the magnitude of $\mathbf{a}+\mathbf{b}$ more than that of $\mathbf{a}-\mathbf{b}$?

(b) When is the magnitude of $\mathbf{a}+\mathbf{b}$ less than that of $\mathbf{a}-\mathbf{b}$?

(c) When is the magnitude of $\mathbf{a}+\mathbf{b}$ equal to that of $\mathbf{a}-\mathbf{b}$?

(d) When is |$\mathbf{a}+\mathbf{b}$| = $|\mathbf{a}|-|\mathbf{b}|$ ?

(e) What is the minimum value of $|\mathbf{a}+\mathbf{b}|$?

Answer 1

Hint:

$$|\mathbf{a}+\mathbf{b}|^2=a^2+b^2+2\mathbf{a} \cdot \mathbf{b}$$

$$|\mathbf{a}-\mathbf{b}|^2=a^2+b^2-2\mathbf{a} \cdot \mathbf{b}$$