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3D vector addition / algebra

I have 3 point charges, each with a 3D vector notation.

A is at (0, 2a, 0)

B is at (3a/2, 0, 0)

C is at (-3a/2, 0,0)

I'm usng Coulombs Law (not gone into here) & the working I have been given is;

F = (1/ 4*pi*E_{0}) * (q^{2} / [r_{A} - r_{B}])^{3} (r_{A} - r_{B})

F = (1/ 4*pi*E_{0}) * (q^{2} / [(-3a/2, 2a, 0)]^{3}) * (-3a/2, 2a, 0)

Which is OK for me, standard, really. It's the next bit which I don't get how to get to, with the a moving from everywhere to one place.

F = (q2 / 4*pi*E_{0*}a^{2}) * (-3/2, 2, 0)/(9/4 + 4)^{3/2}

Could someone please just go through it. I'm sure I'm missing something elementary.

I included a pic of the question if that makes it easier to see.

Re: 3D vector addition / algebra

Quote:

Originally Posted by

**froodles01** I have 3 point charges, each with a 3D vector notation.

A is at (0, 2a, 0)

B is at (3a/2, 0, 0)

C is at (-3a/2, 0,0)

I'm usng Coulombs Law (not gone into here) & the working I have been given is;

F = (1/ 4*pi*E_{0}) * (q^{2} / [r_{A} - r_{B}])^{3} (r_{A} - r_{B})

F = (1/ 4*pi*E_{0}) * (q^{2} / [(-3a/2, 2a, 0)]^{3}) * (-3a/2, 2a, 0)

Which is OK for me, standard, really. It's the next bit which I don't get how to get to, with the a moving from everywhere to one place.

F = (q2 / 4*pi*E_{0*}a^{2}) * (-3/2, 2, 0)/(9/4 + 4)^{3/2}

Could someone please just go through it. I'm sure I'm missing something elementary.

I included a pic of the question if that makes it easier to see.

1.$\displaystyle \langle \frac32 a, 2a, 0 \rangle = a \cdot \langle \frac32, 2, 0 \rangle$

2. I assume that

$\displaystyle \left[ \langle \frac32 a, 2a, 0 \rangle \right]$

is used to caculate the absolute value of $\displaystyle \langle \frac32 a, 2a, 0 \rangle$

If so:

$\displaystyle \left[ \langle \frac32 a, 2a, 0 \rangle \right]^3 = a^3 \cdot \left[ \langle \frac32 , 2, 0 \rangle \right]^3$

Now cancel the factor a in the numerator and the denominator which will leave you with $\displaystyle a^2$ in the denominator.

3. The absolute value of $\displaystyle || \langle \frac32, 2, 0 \rangle || = \sqrt{\langle \frac32, 2, 0 \rangle \cdot \langle \frac32, 2, 0 \rangle} = \sqrt{\frac94 +4} = \left( \frac94 +4 \right)^\frac12$

which can be simplified to $\displaystyle \frac52$