Hello all. Need to verify some answers to questions i've completed.
Given the vectors v = (1,0,1) and w = (-1,1,2)
i) Find the vector projections of w onto v.
My solution: I used the formula (w . unit vector of v) * unit vector of v
unit vector of v is equal to 1 / (root 2) * (1,0,1)
so then (-1,1,2) . (1/ (root 2), 0, 1 / (root 2)) = ( 1/ (root 2) + 0 + 2 / (root 2)) = 1 / (root 2)
I multiplied 1 / (root 2) by the unit vector of v to get (1/2, 0, 1/2) as the vector projection of w onto v.
ii) Find the projection of w orthagonal to v
My solution: v perpendicular = w - projection of w onto v = (-1,1,2) - (1/2, 0, 1/2) which gave me an answer of (-3/2, 1, 1/2).
iii) Find the area of the parallelogram spanned by the vectors v and w. (Note: this question means to ask the area of the parallelogram whose sides are given by vectors v and w)
My Solution: For this question i simply found the lengths of v (root 6) and w (root 2) and multiplied them together to get an answer of 2 (root 3).
2.) Sketch 4y^2 + 9x^2 + 16y - 18x - 11 = 0
My solution: I won't show the full working because it is too long, but i used basic algebra and completing the square to yield: (y+4)^2 / 9 + (x-1)^2 / 4 = 1
this is the basic form of an elipse graph so,
I set x = +1 and got y = -1, -7 (the y-vertices of the elipse)
and I set y = -4 and got x = 3, -1 (the x-vertices of the ellipse)
The centre of the elipse is at (1,-4)
a is greater than b so the major axis is parallel to x-axis.