# Question #0796e

Feb 4, 2018

The magnitude of the vector is between $1$ and $7$.

#### Explanation:

We don't know the angles that the vectors are at, so we can't find the resultant magnitude.

However, we CAN find a range of possible magnitudes.

If we add two vectors $\vec{a}$ and $\vec{b}$ together, then the range for the new vector's magnitude is:

$\left\mid \left\mid \vec{a} \right\mid - \left\mid \vec{b} \right\mid \right\mid \text{ " le" " abs(veca+vecb)" " le" } \left\mid \left\mid \vec{a} \right\mid + \left\mid \vec{b} \right\mid \right\mid$

This notation may look confusing, but just remember that when the || symbols are around a vector, it means magnitude, and when they're around a scalar, it means absolute value.

So all that this equation is really saying is that the magnitude of the resultant vector must be between:

What you get when the two vectors point in opposite directions

AND

What you get when the two vectors point in the same direction

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using this range, and the fact that $\left\mid \vec{a} \right\mid = 3$ and $\left\mid \vec{b} \right\mid = 4$, we see that:

$\left\mid \left\mid \vec{a} \right\mid - \left\mid \vec{b} \right\mid \right\mid \text{ " le" " abs(veca+vecb)" " le" } \left\mid \left\mid \vec{a} \right\mid + \left\mid \vec{b} \right\mid \right\mid$

$\left\mid 3 - 4 \right\mid \text{ " le" " abs(veca+vecb)" " le" } \left\mid 3 + 4 \right\mid$

$\left\mid - 1 \right\mid \text{ "le" " abs(veca+vecb) " "le" } \left\mid 7 \right\mid$

$1 \text{ " le" " abs(veca+vecb)" " le" } 7$

So we can't determine exactly what the vector's magnitude is based on this problem, but we can say for sure that it's between $1$ and $7$.