Consider the following theorem. If a, b, and c are vectors and c is a scalar, then we have the following properties of the cross product. 1. ax b = −b xa 2. (ca) x b = c(ax b) = ax (cb) 3. ax (b + c) = axb+ axc 4. (a + b) x c = axc + bxc 5. a (bx c) (axb).c . = 6. ax (bx c) = (a c)b - (ab)c Prove the property a x b = -bxa of the given theorem. Let a = (a₁, a₂, a₂) and b = (b₁, b₂, ba). Then we get the following. '1' axb- X - - = (-1) i(a₂b3-a₂b3) -j (a²b₁ — ª₁b²) + k(a₂b₁ — ª₁b₂) X

Q: 4. State with a brief explanation whether the following statements are true or false. (a) The…

A: a) true The vectors (1 ,0) and (0,1) is linearly independent. We know that linearly independent set…

Q: The dot product of two vectors can be written in the form x = 00 s+ X1 X2 and y = in R" is defined…

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Q: For the vectors A = Ux + 3uy + 5uz , B = 2ux, 4uy + 6uz, and C = 3ux + 4uy + 5uz. Show that A x (B x…

A: For the vectors A = Ux + 3uy + 5uz , B = 2ux, 4uy + 6uz, and C = 3ux + 4uy + 5uz. To show :-A x (B x…

Q: QI. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, (0,1,2) and x, =(-3,7,2) are linearly…

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Q: Calculate the scalar products of the following vectors: 10 1 Q-B 5 W1 -4 9 (a) V₁ = (b) V2 = (c) V3…

A: Note: As per rule I need to solve only first three subparts.

Q: Find u · (v x w). This quantity is called the triple scalar product of u, v, and w. u = (4, 4, 4), v…

A: = i(5·0 - 0·(-1)) - j (1·0 - 0·0) + k (1·(-1) - 5·0) = i (0 - 0) - j (0 - 0) + k (-1…

Q: (b) 4x Find any values of x such that the vectors 3 and are parallel X-1 to each other.

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Q: Complete the proof of the remaining property of this theorem by supplying the justification for each…

A: Answer is mentioned below

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Q: 2. Using a = (a₁, a2, a3), b = (b₁,b₂, b3), = (C₁, C₂, C3) and k as scalar, prove all properties of…

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Q: For the vectors A = ux + 3uy + 5uz, B = 2ux + 4uy + 6uz, and C = 3ux + 4uy + 5uz, show that A × ( B…

A: For the given vectors A,B,C show the given property  If A=a1ux +a2uy+a3uz and B= b1ux+b2uy+b3uz then…

Q: . The distance of a point in the 3-D system from the origin a. is defined by the absolute value of…

A: Note: We will answer the first question only as the exact one is not specified. Please resubmit the…

Q: Given the following vectors A = -4a +3a-2a₂ X y B=a_+5a +7a_ X y z C=-2a-4a +6а X y Evaluate |AX…

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Q: Find u ∙ (v × w). This quantity is called the triple scalar product of u, v, and w.u = (2, 0, 1), v…

A: Given, the three vectors u = (2, 0, 1), v = (0, 3, 0), w = (0, 0, 1). And we have to find the triple…

Q: a. A: = a: b: -1 1 1 0 A A This is the result of the inverses of the matrices: 3/2- -1/4 -3/2 5/4 0…

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Q: In a standard Cartesian coordinate system a vector, R, has an x-component of + 5.0 m and a y-…

A: In a standard coordinate system a vector R has an x- component of +5.0 m and a y- component of -4.0…

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Q: Let u = (2, 0), v = (-2, -4), and w = (-4, 1). Find the vector æ that satisfies 4u – v+ x = 9x + w.…

A: Solution:-

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A: Vector is parallel to another iff one is constant multiple of other .example: u=kv Orthogonal when…

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A: Given that u→=3,2v→=-2,-2w⇀=-4,3

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Q: Let = [-5 1] and = [6-6]. Find the following vectors. [ = -6y = [ w = 5x – 6y = [ | v=5z ]]

A: A detailed solution is given below.