Question

1) Let a = 〈4,−5,−2〉 and b = 〈2,−4,−5〉 Find the projection of b onto a. proj a b=

2) Find the area of a triangle PQR, where P=(−2,−4,0), Q=(1,2,−1), and R=(−3,−6,5)

3) Complete the parametric equations of the line through the
points (7,6,-1) and (-4,4,8)

x(t)=7−11

y(t)=

z(t)=

Answer #1

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

1) For vectors a and b find the projection of a on b. a =< 4,
−3 >,b =< 3, 2 >
2) Find the area of the parallelogram with adjacent sides a = i
+ j and b = i + k
3) Find the unit tangent T for the following curve r =< 2 cos
t^2 , 2 sin t^2 >

let A (1 , 2, -3), B (2, 1, 4) and C (0, 0, 2) be three points
in R^3
a) give the parametric equation of the line orthogonal to the
plane containing A, B and C and passing through point A.
b) find the area of the triangle ABC
linear-algebra question
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Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

Find the orthogonal projection of v=[−2,10,−16,−19] onto the
subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

For some positive integers t, y + 5 = (1/t) * (x^2) intersects
x^2 + y^2 = 25 at exactly 3 points P, Q, and R (distinct from one
another). Determine the positive integers t for which the area of
triangle PQR is an integer.

Find the area of the triangle PQR when P = (-1, 3, 1), Q = (0,
5, 2), R = (4, 3, -1).

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

Find the lengths of the sides of the triangle PQR. (a) P(4, −1,
−3), Q(8, 1, 1), R(2, 3, 1)
|PQ| =
|QR| =
|RP| =
(b)
P(5, 1, −1), Q(7, 3,
0), R(7, −3, 3)
|PQ|
=
|QR|
=
|RP|
=

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