I have the problem:
The scalar projection of a vector v on a vector W is the length of v on w.
Now i thought i could just do this:
But then i end up with 24/sqrt(31)
but the answer is supposed to be 24/7
I have also tried going the long way by finding the angle and using the cos(x)=a/c to find the length but i end up with the same wrong answer.
I noticed i had made a small typo.
it was supposed to be w=2i-3j+6k
As for the question itself, my native language is not english so i have tried to translate it. Something might have gotten "lost in translation" I will try to translate another way:
Scalar Projection of a vector v on a vector w is the length of v along w. Make use of the dot product to calculate the Scalar Projection in each case.
But i noticed you used ||w|| insted of |w|. I have not seen that before. Do you mean to take the absolute value of the length of the vector? Isnt that redundant? I thought the length of a vector was always >= 0
look at this.
I know perfectly well what scalar projections are.
Thank you, in my book they have only used single | for both the abs value of the scalar and for the length of a vector. But I see the benefits of using the double || for the length of a vector.
I apologize if I caused any offence. I only repeated the question exactly as it is written in my textbook(see image and link to Google translate)
My apology is not conditional.
the english translation is probably something more along the lines of:
"the (scalar) projection of a vector v on w is the length of v in the direction of w".
the (vector) projection of v in the direction of w is usually defined to be:
we have that:
note that this agrees with Plato's post, up to sign.
if the angle between v and w is less than (or equal to) a quarter-circle, then the two definitions coincide.
if the angle between v and w is more than a quarter-circle, then v points "away" from w, and one uses .
in other words, Plato's definition is more "accurate" (conveys more geometrical information) than what your book apparently says (real numbers can have a sign, as well as a magnitude), and is in fact, the standard definition.
in your original problem (as amended), it makes no difference which formula you use, as the angle between (3,-4,1) and (2,-3,6) is indeed less than a quarter-circle.
so we have that the scalar projection is: (3*2 + (-4)*(-3) + 1*6)/(√[22 + (-3)2 + 62])
= (6 + 12 + 6)/√49 = 24/7
Thank you, you’re explanation has helped me understand the issue a little better Deveno. I had not thought of the sign issue at all.
The reason I posted my question again was because I had tried exactly what Plato suggested and gotten the wrong answer. When you now showed me that it indeed resulted in the right answer I went back over my work and the logs on my calculator and discovered my error.
I had entered sqrt[2^2 + (-3)^2 + 6^2] as sqrt[2^2+-3^2+6^2]. When I went over and did all the boring operations manually I discovered that this caused the (-3)^2 to be calculated as -9 instead of 9.
You learn from your mistakes right?
Thanks both of you.