Checking and Constructing Orthogonal or Parallel Vectors

Determine whether the given vectors are orthogonal, parallel or neither.
<-5,3,7> and <6,-1,2>
<4,6> and <-3,2>
-i + 2j + 5k and 3i + 4j - k
2i + 6j - 4k and -3i -9j +6k

Find a unit vector that is orthogonal to both i+j and i+k.

If a = <3,0,-1>, find a vector b such that comp_a b = 2 (component of b in the a direction is 2).

Please see the attached file. Included in text here as much as possible, but the file is better.
Hi.
In this answer, I'm generally going to change i, j, and k to the appropriate numerical vectors, because it's simpler this way. However, you could do those problems without that, and most likely you'll have to as things go on in the course.

a. We check for orthogonality by taking the dot product of the two vectors. If it's 0, they're orthogonal. Remember that the dot product is the sum of the products of the appropriate component pairs.
a?b = -5*6 + 3*(-8) + 7*2 = -30 - 24 +14 = -40 They are not orthogonal
We check for parallelness by checking if the vectors are scalar multiples of each other (that is, you can multiply one by a specific number to get the other). We can do this by taking pairs of components (for example the -5 and 6 pair) and dividing one by the other, and seeing if we always get the same answer.
-5/6, 3/(-8), and 7/2 : are they the same number? No, certainly not; for example -5/6 is not the same as 7/2.
b.
a?b = 4*(-3) + 6*2 = -12 + 12 = 0
They are orthogonal! This also means that they are not parallel; nonzero orthogonal vectors cannot also be parallel. Note that 4/(-3) is not the same as 6/2.
c. Here I will rewrite the unit vector sums as numerical vectors.
a = -1*i + 2*j + 5*k = -1*<1, 0, 0> + 2*<0, 1, 0> + 5*<0, 0, 1> = <-1, 2, 5>
b = <3, 4, -1>
a?b = -1*3 + 2*4 + 5*(-1) = -3 + 8 -5 = 0
Again the vectors are orthogonal, and therefore not parallel. Note that -1/3 is not equal ...

Solution Summary

This solution shows how to check four different pairs of vectors to see if they are orthogonal, parallel, or neither. It also shows how to construct a unit vector orthogonal to two other vectors, and how to construct a vector with a given component in a given direction.

1) For which values of k are the following vectors u and v orthogonal?
a) u = (2,1,3) , v = (1,7,k)
b) u = (k,k,1) , v = (k,5,6)
2) Let u,v be orthogonal unit vectors. Prove that d(u,v) = 2^(1/2)
(The questions are unrelated)

3. Let V be an R-vector space with inner product ( - . - ).
(a) Let S = {b1, b2, ...} be a set of vectors in V. Define what it means for S to be an orthogonal set or an orthonormal set with respect to the inner product.
(b) Let V = R^4 and let ( - . - ) be the dot product. Apply the Gram-Schmidt orthoganlisation process to t

Consider the following three vectors in R^3:
x_1=(1, -1, 0, 2)
x_2=( 1,1,1,0)
x_3= (-1,-1,2,0)
a) Verify that {x_1, x_2, x_3} are orthogonal with the standard inner product in R^4
b) Find a nonzero vector x_4 such that {x_1, x_2, x_3, x_4} is a set of mutually orthogonalvectors.
c) Convert the resulting set into an ort

Please see the attached file for full problem description.
1.Let u=(1,-1,3) and v=(2,-1,-1) be vectors in Euclidean 3-space R^3. Find a vector orthogonal to the plane of (subspace spanned by) the vectors u and v. Show work.
Help: R^3: is Euclidean 3-space

In C[-pi, pi] with inner product defined by (6), show that cos mx and sin nx are orthogonaland that both are unit vectors. Determine the distance between the two vectors.
(6) (f,g) = (1/pi)* the integral from -pi to +pi of f(x)g(x)dx
This is all from Linear Algebra With Applications by Steven J. Leon, Sixth Edition. Than

Given a vector w, the inner product of R^n is defined by:
=Summation from i=1 to n (xi,yi,wi)
[a] Using this equation with weight vector w=(1/4,1/2,1/4)^t to define an inner product for R^3 and let x=(1,1,1)^T and y=(-5,1,3)^T
Show that x and y are orthogonal with respect to this inner product. Compute the values of

1. For vectors v and w in , show that v - w and v + w are perpendicular if and only if .
2. Let u = (-3, 1, 2), v = (4, 0, -8), and w = (6, -1, -4) be vectors in . Find the components of the vector x that satisfies 2u - v + x = 7x + w.
3. Find a non-zero u vector such that satisfies the following.
a. u has the same d

How to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1