1. For this question, we define the following vectors: u = (1, 2), v = (−2,...

1. For this question, we define the following vectors: u = (1, 2), v = (−2, 3).

(a) Sketch following vectors on the same set of axes. Make sure to label your axes with a scale. i. 2u ii. −v iii. u + 2v iv. A unit vector which is parallel to v

(b) Let w be the vector satisfying u + v + w = 0 (0 is the zero vector). Draw a diagram showing the geometric relationship between the three vectors u, v and w.

2. Let P 1 and P2 be planes with general equations P1 : −2x + y − 4z = 2, P2 : x + 2y = 7.

(a) Let P3 be a plane which is orthogonal to both P1 and P2. If such a plane P3 exists, give a possible general equation for it. Otherwise, explain why it is not possible to find such a plane. (b) Let ` be a line which is orthogonal to both P1 and P2. If such a line ` exists, give a possible vector equation for it. Otherwise, explain why it is not possible to find such a line.

Expert Solution

Colby Messinger answered 2 years ago

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ =...

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉. Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

show that for any two vectors u and v in an inner product space ||u+v||^2+||u-v||^2=2(||u||^2+||v||^2) give...

show that for any two vectors u and v in an inner product space ||u+v||^2+||u-v||^2=2(||u||^2+||v||^2) give a geometric interpretation of this result fot he vector space R^2

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 . We wish to...

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 . We wish to prove that -> (u · v)^ 2 ≤ |u|^ 2 |v|^2 . This inequality is called the Cauchy-Schwarz inequality and is one of the most important inequalities in linear algebra. One way to do this to use the angle relation of the dot product (do it!). Another way is a bit longer, but can be considered an application of optimization. First, assume that the...

For the following exercises, use the vectors shown to sketch u + v, u − v, and 2u.

Let u and v be vectors in R3. Consider the following statements.T or F (1) |u ...

Let u and v be vectors in R3. Consider the following statements.T or F (1) |u · v| ≤ ||u|| + ||v|| (2) If au + bv = cu + dv then a = c and b = d. (3) ||u + v||2 = ||u||2 + ||v||2 + 2(u · v) Let u, v, and w be vectors in R3. T or F. (1) u · v − ||u|| (2) (u · v) × w (3) || ( ||u|| projvu ...

For the following exercises, calculate u ⋅ v. Given the vectors shown in Figure 4, sketch u + v, u − v and 3v.

For the following exercises, calculate u ⋅ v.Given the vectors shown in Figure 4, sketch u + v, u − v and 3v.

1) Determine the angle between vectors: U = <2, -3, 4> and V= <-1, 3, -2>...

1) Determine the angle between vectors: U = <2, -3, 4> and V= <-1, 3, -2> 2) determine the distance between line and point P: -2x+3y-4z =2 L: 3x – 5y+z =1 3) Determine the distance between the line L and the point A given by L; (x-1)/2 = (y+2)/5 = (z-3)/4 and A (1, -1,1) 4) Find an equation of the line given by the points A, B and C. A (2, -1,0), B (-2,4,-1) and C ( 3,-4,1)...

2 Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and w=(3,5,4,4). 2.1 Construct a basis...

2 Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and w=(3,5,4,4). 2.1 Construct a basis for the vector space spanned by u, v and w. 2.2 Show that c=(1,3,2,1) is not in the vector space spanned by the above vectors u,v and w. 2.3 Show that d=(4,9,17,-11) is in the vector space spanned by the above vectors u,v and w, by expressing d as a linear combination of u,v and w.

All vectors are in R^ n. Prove the following statements. a) v·v=||v||2 b) If ||u||2 +...

All vectors are in R^ n. Prove the following statements. a) v·v=||v||2 b) If ||u||2 + ||v||2 = ||u + v||2, then u and v are orthogonal. c) (Schwarz inequality) |v · w| ≤ ||v||||w||.

Let vectors u and v form a basis in some plane, in each of the following...

Let vectors u and v form a basis in some plane, in each of the following cases determine if the vectors e1 and e2 for a basis in this plane: a)e1=u+v e2=u-v b)e1=-u+2v e2=3u-6v substantiate your decision

Question