I have two vectors:
$A: \{51.031, -102.062, 51.031\}\ (|A|= 125)$
$B: \{2, -4, -1\}\ (|B| =\sqrt{21}\approx 4.58)$
I am trying to find the amount of $\vec{A}$ in the $\vec{B}$ direction, so I used dot product with the coordinate method:
$(x_1 \times x_2+y_1 \times y_2+z_1 \times z_2)$
$([2 \times 51.031]+ [-4 \times -102.062]+ [-1 \times 51.031]) = 459.279$
If the dot product is supposed to find the magnitude of a vector that is pointing in the direction of another vector, how do I get a result that's more than $3$ times the length of the longest vector?
What am I doing wrong?
The magnitude of $A$ in $B$'s direction (i.e. the magnitude of the projection of $A$ onto $B$) is given by $\frac{A\cdot B}{|B|},$ not $A\cdot B.$
You can see that $A\cdot B$ is wrong by units, too, since it has the units of $A$ times $B,$ rather than $A.$ If we situate $B$ on the $x$-axis, we want $|A|\cos\theta,$ where $\theta$ is the angle between $A$ and $B.$ Recall that the dot product is given by $A\cdot B=|A||B|\cos\theta,$ so $\frac{A\cdot B}{|B|}=|A|\cos\theta,$ which is what we want.