Hi @Mikapo, your responses are easy enough to read.
You could opt to include more of the intermediate steps to make it easier to figure if/where an you make a mistake (in case you do end up with the wrong answer), but overall I don’t have any problems following the calculations that you’re doing.
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the is (103/147, 94/147, -2/147)
I have been trying to understand the formula for the last two days, for the time being I will just try and do the maths
-2 * (2/21 - 12/21 - 6/21) . (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3) # here I am switching initial 2 sign to make things easier to type
(32/21 * 2/7, 32/21 * 6/7, 32/21 * 3/7) + (1/3, -2/3, -2/3)
(64/147, 64/49, 32/49) + (1/3, -2/3, -2/3)
(113/147, 94/147, -2/147)
I am stuck at one minor point Gary. In post #3 you go from 2(-(1/3, -2/3, -2/3) • (2/7, 6/7, 3/7)) to 2(16/21).
And I can not wrap my head around that. I get the denominator of 21, I got the same. But in my calculations I end up at 672/21. And not 32/21.
It looks like that 2((-42/21) + 252/21 + 126/21)
It’s probably pre-school math operations I am lacking here. But I simply can’t solve it on my own. Thanks for the help.
For this step we’re taking the dot product of two vector, so we just need to multiply each element of the two vectors and then add them. Remember that when we multiply fractions we multiply both the numerator and denominator.
Here are the expanded steps for you:
-(1/3, -2/3, -2/3) • (2/7, 6/7, 3/7)
= (-1/3, 2/3, 2/3) • (2/7, 6/7, 3/7) <- eliminate the leading negative on the first vector
= ((-1*2)/(3*7), (2*6)/(3*7), (2*3)/(3*7)) <- multiply the numerators AND denominators of each component
= (-2/21, 12/21, 6/21)
= (-2/21 + 12/21 + 6/21) <- add each component to get the scalar product
= (-2 + 12 + 6)/21
= 16/21
I hope that clears things up for you.
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Ah yes! I get it now. Thanks @garypettie! My mistake was the following in the case of the x-vectors (and all other subsequently):
Instead of calculating…
(-1*2)/(3*7) = -2/21
… I thought that I first have to find the common denominator of 21 by multiplying
-1/3 * 7 = -7/21
and
2/7 * 3 = 6/21
and after that I did multiply the two (numerators). Which gave me
-42/21 (Which added up nicely to 2 I might add)
See, I told it was some really basic math operations I was lacking. (/me goes into the corner of shame. :D) Back to the drawing board.
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Don’t worry, it happens to the best of us!
It sounds like you partially confused adding and multiplying fractions.
When we add fractions, we have to find the common denominator and then add the numerators.
For multiplication we just straight multiply both the numerator and denominator.
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Ok, with my newly-acquired super power of multiplying fractures I came to the following conclusion:
r = 2((-1/3 * 2/7) + (2/3 * 6/7) + (2/3 * 3/7))n + a
r = 2(-2/21 + 12/21 + 6/21)n + a
r = 32/21(2/7, 6/7, 3/7) + a
r = (64/147, 192/147, 96/147) + (1/3, -2/3, -2/3)
r = (64/147, 192/147, 96/147) + (49/147, -98/147, -98/147)
r = (113/147, 94/147, -2/147)
(Writing this stuff correctly is a challenge unto itself.)
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The answer is (0.7,0.58,-0,01)
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r = 2(-0.330.29 + 0.660.86 + 0.660.43) * (0.29, 0.86, 0.43) + (0.33, -0.66, -0.66)
r = 1.511 (0.29, 0.86, 0.43) + (0.33, -0.66, -0.66)
r = (0.44, 1.3, 0.65) + (0.33, -0.66, -0.66)
r = (0.77, 0.64, -0.01)
My answer:(0.77,0.64,-0.01)
this was very hard, probably wrong
.6704,.5655 ,.194
i used extra decimal spaces for repeating numbers.
r = 2(-a∙n) x n + a
n = (2/7, 6/7, 3/7)
a = (1/3, -2/3, -2/3)
-a∙n = (-1/3) * (2/7) + (2/3) * (6/7) + (2/3) * (3/7) = 16/21
2(-a∙n) = 2 * 16/21 = 32/21
2(-a∙n) x n = (32/21) * (2/7, 6/7, 3/7)
= (32/21 * 2/7), (32/21 * 6/7), (32/21 * 3/7)
= (64/147, 192/147, 96/147)
2(-a∙n) x n + a = (64/147, 192/147, 96/147) + (1/3, -2/3, -2/3)
= (64/147, 192/147, 96/147) + (49/147, -98/147, -98/147)
= (113/147, 94/147, -2/147)
r = (0.77, 0.64, -0.01)
R = 2 (-A • N) x N + A
R = 2 ((-0,33 * 0,29) + (0,66 * 0,86) + (0,66 * 0,43)) x N + A
R = 1,51 x (0,29; 0,86; 0,43) + (0,33; -0,66; -0,66)
R = (0,44; 1,30; 0,65) + (0,33; -0,66; -0,66)
R = (0,77; 0,64; -0,01)
r = 2(â · n̂) * n̂ + â
r = 2((-0.33 * 0.29) + (0.66 * 0.86) + (0.66 * 0.43)) * n̂ + â
r = 1.51 (0.29, 0.86, 0.43) + (0.33, -0.66, -0.66)
r = (0.44, 1.3, 0.65) + (0.33, -0.66, -0.66)
r = (0.77, 0.64, -0.01)
Here’s my work for the Reflection challenge:
Let N be our normalized normal vector
Let A be our normalized starting vector
Let R be our normalized reflection vector
R = 2((-1/3, 2/3, 2/3) * (2/7, 6/7, 3/7)) x (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
R = 2(16/21) x (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
R = 32/21 x (2/7, 6/7, 3/7) + (1/3, -2/3, -2/3)
R = (64/147, 192/147, 96/147) + (1/3, -2/3, -2/3)
R = (113/147/ 94/47, -2/147)
or expressed in decimal format…
R = (0.77, 0.64, -0.01)
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Hah! Very nervous about my attempt. But, here is what I’ve got scribbled down in my notepad. Sorry I don’t know how to insert fancy vector formatting characters into my text!
r = 2((-1((0.3, -0.6, -0.6))) · (0.29, 0.86, 0.43)) x (0.29, 0.86, 0.43) + (0.3, -0.6, -0.6)
Worked DOT Product first…
r = 2((-0.087, 0.516, 0.258)) x (0.29, 0.86, 0.43) + (0.3, -0.6, -0.6)
r = 2(0.687) x (0.29, 0.86, 0.43) + (0.3, -0.6, -0.6)
r = 1.374(0.29, 0.86, 0.43) + (0.3, -0.6, -0.6)
Scalar, rounding to nearest hundredth.
r = (0.4, 1.18, 0.59) + (0.3, -0.6, -0.6)
r = (0.7, 0.58, -0.01)
Looking through other answers after I submitted mine, this is the only one that matches what I found. Makes me worry! LOL Hope we find out what the answer was in the next lesson!
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Sorry I’m trying to wrap my head around this concept. Where did you get (-0.33, 0.66, 0.66) on the second line?