the symbol$$A _{b}$$ stands for the projection of vector A onto vector B. In other words, $$A _{b}$$ represents the component of A that is parallel to B. Derive an expression for $$A _{b}$$ in terms of the vectors A and B.

Question

the symbol$$A _{b}$$ stands for the projection of vector A onto vector B.  In other words, $$A _{b}$$ represents the component of A that is parallel to B.  Derive an expression for $$A _{b}$$ in terms of the vectors A and B.

amistre64 · Answer

$$\frac{a.b}{|a|^2}a$$

anonymous · Answer

wow u did that really quickly

amistre64 · Answer

just got out of calc3 that just went over it :)

anonymous · Answer

in the numerator...is that a times b?

amistre64 · Answer

a dot b

anonymous · Answer

a dot b divided by magnitude of a squared all multiplied by a?

amistre64 · Answer

yes, the left side is a scalar; and the right side is vector a scaled to that length

anonymous · Answer

would it be a bother to run me through how u got to this expression?

amistre64 · Answer

|dw:1327527260794:dw|

amistre64 · Answer

|b| cos(t) = the length of Ab, if we use your notation for proj{a} b

amistre64 · Answer

$$|b|cos(t) =|b| \frac{a.b}{|a||b|}=\frac{a.b}{|a|}$$
good so far?

anonymous · Answer

yea i get that

anonymous · Answer

i'm with u

anonymous · Answer

i'm also trying to decode your picture :)

anonymous · Answer

is that the x coordinate down there at the end?

anonymous · Answer

the long vector

amistre64 · Answer

now, we need to scale that to a unit vector of a; since a is |a| long, lets divide it by |a| to get it to a length of 1

amistre64 · Answer

$$\frac{a.b}{|a|}\ \frac{a}{|a|}=; unit\ a, scaled\ by\ needed\ length$$

anonymous · Answer

okay

amistre64 · Answer

and since |a| |a| = |a|^2 i just condensed it alittle bit

anonymous · Answer

simple enough?

amistre64 · Answer

|dw:1327527605851:dw|