given v=5i+3j and w=-4i+8j, find projwv
There are two kinds of projection of w on to v or of v on to w.
You did not say projection of which on the other or if you want scalar or vector projection.
see below:
Projections
One important use of dot products is in projections. The scalar projection of b onto a is the length of the segment AB shown in the figure below. The vector projection of b onto a is the vector with this length that begins at the point A points in the same direction (or opposite direction if the scalar projection is negative) as a.
Thus, mathematically, the scalar projection of b onto a is |b|cos(theta) (where theta is the angle between a and b) which from (*) is given by
This quantity is also called the component of b in the a direction (hence the notation comp). And, the vector projection is merely the unit vector a/|a| times the scalar projection of b onto a:
Thus, the scalar projection of b onto a is the magnitude of the vector projection of b onto a.
Here:
http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html
it was proj with w as a subscript of proj and then v
I am sorry. I am not familiar with that notation. Follow the recipe in the link I gave you.
To find the projection of vector v onto vector w, you can use the formula:
projwv = (v · ŵ) * ŵ
where v · ŵ represents the dot product of vectors v and ŵ, and ŵ is the unit vector in the direction of w.
Let's break down the calculation step by step:
Step 1: Find the unit vector ŵ
To find the unit vector ŵ, we need to normalize vector w by dividing it by its magnitude (length).
The magnitude of w is given by:
|w| = √((-4)^2 + 8^2) = √(16 + 64) = √80 = 4√5
Dividing w by its magnitude, we get:
ŵ = (-4i + 8j) / (4√5) = (-i + 2j) / √5
Step 2: Calculate the dot product (v · ŵ)
The dot product of v (5i + 3j) and ŵ (-i + 2j) can be calculated by taking the product of their corresponding components and summing them up.
(v · ŵ) = (5*(-1) + 3*2) = (-5 + 6) = 1
Step 3: Multiply (v · ŵ) by ŵ
Now, we multiply the dot product (v · ŵ) by ŵ to find the projection of vector v onto vector w.
projwv = (v · ŵ) * ŵ = 1 * (-i + 2j) / √5 = -i/√5 + 2j/√5
Therefore, the projection of vector v onto vector w is -i/√5 + 2j/√5.