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1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5,5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel /ABC; (3) Find the area of AABC.

1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5,5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel /ABC; (3) Find the area of AABC.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 45E
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1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5, 5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel LABC; (3) Find the area of AABC. (b) Find a vector equation for the line that passes through A and B; also find the distance from C to the line. (c) Find a linear equation for the plane that passes through A,B, and C; also find a point D on the plane such that OD is perpendicular to the plane. 2. Linear Equations (a) Let k be a constant. For unknown x, y, z, solve the system x+y+z = 1, 2x+y+z = 5,6x + y + z = k. (b) Let k be a constant. For unknown x, y, z, solve the system x + y + kz = 1, x + ky + z = 1, kx+y+z= −2. (c) Show that u, v, w is in span{u + v, 2u + 3v, 4v +6w}. (d) Check the linear dependency of the matrices B,C,D in the next problem. = (e) Let V₁ = = [1 2 3], V₂ V1 and v₂. [2 5 6], u = [210] and w = [1 1 1]. Is u E span(V₁, V₂)? Is we span (v₁, V2)? If yes, write u or w as a linear combination of 3. Matrix Algebra (a) Let A = 1 2 3 45 " (b) Let A = B = [43] 1 3 1 -1 0 1 2 1 -2 1 3 1 6 1 с -3 4 also find rank(A) and nullity(A). 6 7 = - [83] P-[88] D = 9 E Find (1) AAT and ATA; (2) Find B-¹ and I - 6B – 3B²; (2) Solve matrix X from XBCX = D. (3) Find E-¹. 1 = 1 2 22 4 1 3 -3 Find bases for row(A), col(A), and null(A);
Transcribed Image Text:1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5, 5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel LABC; (3) Find the area of AABC. (b) Find a vector equation for the line that passes through A and B; also find the distance from C to the line. (c) Find a linear equation for the plane that passes through A,B, and C; also find a point D on the plane such that OD is perpendicular to the plane. 2. Linear Equations (a) Let k be a constant. For unknown x, y, z, solve the system x+y+z = 1, 2x+y+z = 5,6x + y + z = k. (b) Let k be a constant. For unknown x, y, z, solve the system x + y + kz = 1, x + ky + z = 1, kx+y+z= −2. (c) Show that u, v, w is in span{u + v, 2u + 3v, 4v +6w}. (d) Check the linear dependency of the matrices B,C,D in the next problem. = (e) Let V₁ = = [1 2 3], V₂ V1 and v₂. [2 5 6], u = [210] and w = [1 1 1]. Is u E span(V₁, V₂)? Is we span (v₁, V2)? If yes, write u or w as a linear combination of 3. Matrix Algebra (a) Let A = 1 2 3 45 " (b) Let A = B = [43] 1 3 1 -1 0 1 2 1 -2 1 3 1 6 1 с -3 4 also find rank(A) and nullity(A). 6 7 = - [83] P-[88] D = 9 E Find (1) AAT and ATA; (2) Find B-¹ and I - 6B – 3B²; (2) Solve matrix X from XBCX = D. (3) Find E-¹. 1 = 1 2 22 4 1 3 -3 Find bases for row(A), col(A), and null(A);
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