Question

Where does the pi/2 comes from?

Answer 1

Where do they get the pi/2 after the cos? I must be missing something

Answer 2

good question. They pulled it out of their hat (or at least, did not explain how they got it)

you can find the slope from 0,0 to 1,1, i..e 1
and the slope of the line from 0,0 to -3,3 i.e. -1
the slopes are "negative reciprocals"
\[ m_1 = - \frac{1}{m_2} \]
and that means the lines are perpendicular, and form a 90º angle, or \( \frac{\pi}{2}\) radians

Answer 3

Ah, interesting. Because for the question that comes right after the lesson goes like this...
The angle between vectors (2, 3) and (-3, 2) is _ degrees.

Would I calculate the slope to find the angle?

Answer 4

no. we use the two different ways to compute the dot product
the "easy" way (at least for me) is multiply corresponding components and add
the other way is
v dot w = |v| |w| cos x
we know v dot w (from the first way)
we know the lengths (or magnitudes) of v and w
that means we can "solve for cos x"

Answer 5

The dot product will be zero, right? so then I plug that into this:
v dot w = |v| |w| cos x
what do the v and w values represent?

Answer 6

I just mean two random vectors v and w
the | v | means the length of v (the square root of the sum of the squares of the components)

here you know the dot product is 0
so you have
0 = |(2, 3)| | (-3, 2) | cos x
we could figure out the actual lengths (both are sqr(8+4)= sqr(12)= 2sqr(3) )
but you will end up with
cos x = 0
and
\[ x = \cos^{-1}0 = 90º\]

Answer 7

**
we could figure out the actual lengths (both are sqr(9+4)= sqr(13)

Answer 8

|(2, 3)| | (-3, 2) | does this mean we add the vectors?

Answer 9

no. the | v | means "find the length of vector v"

for example, for the vector (1,1,)
we could do this:

Answer 10

for the vector (2,3)
(2 over, 3 up)
we use
c^2= a^2 + b^2 where a is 2, b is 3
c^2 = 2^2 + 3^2
c^2= 4+9
c^2= 13
c= sqr(13)

that is using Pythagorean Theorem

but people just square the components:
(2,3) --> do 2*2 + 3*3 = 4+9 = 13
then take the square root : sqr(13)
that is the "length" of the vector.

Answer 11

the formula
\[ v \cdot w =\ |v|\ |w| \cos \theta\]
means multiply the length of vector v times the length of vector w
then multiply by the angle between the two vectors (assuming you know it)

Answer 12

Oh! So my equation would look like this:
0 = 2sqrt13costheta
And solve for theta?