Establish the following vector inequalities geometrically or otherwise:
$(a) \quad |\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$
$(b) \quad |\vec{a} + \vec{b}| \geq ||\vec{a}| - |\vec{b}||$
$(c) \quad |\vec{a} - \vec{b}| \leq |\vec{a}| + |\vec{b}|$
$(d) \quad |\vec{a} - \vec{b}| \geq ||\vec{a}| - |\vec{b}||$
When does the equality sign apply in each case?

(N/A) Let two vectors $\vec{a}$ and $\vec{b}$ be represented by the adjacent sides of a parallelogram $OMNP$. In $\Delta OMN$,the side $ON$ represents $\vec{a} + \vec{b}$. By the triangle inequality,the length of any side is less than or equal to the sum of the lengths of the other two sides: $|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$. Equality holds when $\vec{a}$ and $\vec{b}$ are in the same direction.
$(b)$ In $\Delta OMN$,the difference of two sides is less than or equal to the third side: $|\vec{a} + \vec{b}| \geq ||\vec{a}| - |\vec{b}||$. Equality holds when $\vec{a}$ and $\vec{b}$ are in the same direction.
$(c)$ Similarly,for the vector difference $\vec{a} - \vec{b}$,using the triangle formed by $\vec{a}$ and $-\vec{b}$,we get $|\vec{a} - \vec{b}| \leq |\vec{a}| + |-\vec{b}| = |\vec{a}| + |\vec{b}|$. Equality holds when $\vec{a}$ and $\vec{b}$ are in opposite directions.
$(d)$ Using the triangle inequality for the difference,we have $|\vec{a} - \vec{b}| \geq ||\vec{a}| - |\vec{b}||$. Equality holds when $\vec{a}$ and $\vec{b}$ are in opposite directions.