Vectors

**strgrl** · April 4th 2009, 12:39 PM

Hi, does anyone know if there is/what the general formula for the addition of vectors is?

In particular, I'm trying to work out the problem: Two vectors

and

have precisely equal magnitudes. In order for the magnitude of

+

to be larger than the magnitude of

-

by the factor n, what must be the angle between them?

A friend offered the formula: sum^2 = a^2 + a^2 + a^2cos(t)
diff^2 = a^2 + a^2 - a^2cos(t)

However, I've never seen these formulas before and I don't understand the premise behind them so cannot determine if they are correct...

Can someone please help explain these formulas? Or offer another method of solving this problem?

Thank you!

**Plato** · April 4th 2009, 01:15 PM

Originally Posted by strgrl

Two vectors

and

have precisely equal magnitudes. In order for the magnitude of

+

to be larger than the magnitude of

-

by the factor n, what must be the angle between them?

I find this question a bit vague.
We assume that from the wording that $n >1$ .
In which case in order for $\left\| {A + B} \right\|$ to be greater than $\left\| {A - B} \right\|$ the angle between $A~\&~B$ must be acute.
(The length of $A~\&~B$ is irrelevant to the problem.)

Look at what follows.
$\left\| {A + B} \right\|^2 = \left\| A \right\|^2 + 2A \cdot B + \left\| B \right\|^2$
$\left\| {A - B} \right\|^2 = \left\| A \right\|^2 - 2A \cdot B + \left\| B \right\|^2$
So in order for $\left\| {A + B} \right\| > \left\| {A - B} \right\|$ it must be the case that $A \cdot B > 0$ .
That happens only if the angle between them is acute.

**Soroban** · April 4th 2009, 02:05 PM

[size=3]Hello, strgrl!

Two vectors $\vec A \text{ and }\vec B$ have equal magnitudes.
In order for the magnitude of $\vec A + \vec B$ to be larger than the magnitude of $\vec A - \vec B$
by the factor $n$ , what must be the angle between them?

A friend offered the formula: . $\begin{array}{ccc}\text{sum}^2 &=& a^2 + a^2 + a^2\cos\theta \\\text{diff}^2 &=& a^2 + a^2 - a^2\cos\theta \end{array}$

However, I've never seen these formulas before
and I don't understand the premise behind them
so cannot determine if they are correct...

Can someone please help explain these formulas?
Or offer another method of solving this problem?

I believe your friend is using the Law of Cosines ... but incorrectly.

The sum of the vectors, $\vec A + \vec B$ , is the diagonal of their parallelogram.

Code:

            .  .  .  .  .  .  *
           .              *  *
          .    A+B    *     *
         .        *        * B
        .     *           *
       .  *            θ *
      *  *  *  *  *  *  *
               A

Code:

            *  .  .  .  .  .  .
           *  *              .
          *     *  A-B      .
      -B *        *        .
        *           *     .
       * 180°-θ       *  .
      *  *  *  *  *  *  *
               A

Law of Cosines: . $|A + (\text{-}B)|^2 \;=\;|A|^2 + |\text{-}B|^2 - 2\!\cdot\!|A|\!\cdot\!|\text{-}B|\cos(180^o -\theta)$

. . which simplifies to: . $|A - B|^2\;=\;|A|^2 + |B|^2 + 2\!\cdot\!|A|\!\cdot\!|B|\cos\theta$

Your turn . . .

**Plato** · April 4th 2009, 03:53 PM

Originally Posted by Soroban

The sum of the vectors, $\vec A + \vec B$ , is the diagonal of their parallelogram.

Code:

            .  .  .  .  .  .  *
           .              *  *
          .    A+B    *     *
         .        *        * B
        .     *           *
       .  *            θ *
      *  *  *  *  *  *  *
               A

If you look carefully at your own diagram, the diagonal you have described is $B-A$ and not $A+B$

**mr fantastic** · April 5th 2009, 03:21 AM

Originally Posted by Plato

If you look carefully at your own diagram, the diagonal you have described is $B-A$ and not $A+B$

Without arrow heads I suppose there is a certain ambiguity. However, I think all the posts in this thread collectively give enough help for the OP to work out the answer to the question.

Thread: Vectors

LinkBack

Thread Tools

Display

Vectors

Similar Math Help Forum Discussions

Calculus 3 Questions: Tangent Vectors/Normal Vectors, Arc Length and Curvature?

[SOLVED] Why row vectors equal to linear combination of basis vectors of rowspc(in this case)?

[SOLVED] Row vectors of B and the subspace spanned by row vectors of B lie in row space A:Why?

[SOLVED] Vectors: Finding coefficients to scalars with given vectors.

A matrix that permutates row vectors to column vectors

Search Tags