Show that r|(1) and r2(1) define the same line, where rị(t) = (3, –1, 4) + t (8, 12, –6) r2(1) = (11, 11, –2) + 1 (4, 6, –3) Hint: Show that r2(1) passes through (3, – 1, 4) and that the direction vectors for r¡(1) and r2(1) are parallel.

Q: 4- Determine whether the vectors v1 = (1, –1, 2), v2 = (3, 2, – 1) and v3 = (5,0, 3) form a linearly…

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Q: Suppose , t2, ., , are linearly independent. Prove that S is linearly independent where (a) S=…

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Q: (c) If f is a C² scalar function, then V × (Vƒ) = 0. True False (d) The lines 1₁ (t) = (1 − t, 1+ t,…

A: (C) If c is scalar function Means      curl(grad ƒ)  is = 0   Therefore correct ans is  True

Q: 3. (а) Consider the vectors ū = i – 6j + 4k, ở = 2i – 5j – 3k and w = 6i – 2j + 3k Show that ū x (T…

A: Use the formula for determining the cross product of two vectors  Substitute the values given and…

Q: (a) [1 2]. [–1 1] (c) [1 3], [2 – 3]. [0 2] (d) [2 4], [-1 2] (b) [0 0]. [1 1]. [–2 – 2]

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Q: (c) Determine the values of a and b if T(x,y,z) = (2x+y+z,a+3y,b+4z) is not self- adjoint.

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Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Verify that x − p is orthogonal to p.

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Q: Let L₂ be the line passing through the point P₁=(-3, -1, -2) with direction vector -[0, 3, 3], and…

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Q: Which of the following is a linear combination of the vectors v₁ = (a) (²) (b) (-₂²) (c) (4) (-¹)…

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Q: 5. Determine whether or not the following vectors i R³ are linearly dependent: (1, –2, 1), (2, 1,…

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Q: Use the function to find the image of v and the preimage of w. T(V1, V2) = (V1 + V2, V1 - V½), v =…

A: To solve this problem we follow the following steps : Finding image :  Write down the given…

Q: Find the vector ž, which is perpendicular to vectors å = (2,3, –1) and 5 = (1,–2,3) and satisfies…

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Q: If a, and b are non - collinear vector and vectors a = (x +4 y)a +(2x + y +1)b and, B= (-2x + y + 2…

A: We will solve the following

Q: Use the function to find the image of v and the preimage of w. T(V1, v2) = (Vị + v2, V1 - V2), v =…

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Q: or function By – k1 z) + ay (kz y + z) onal, then the values

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Q: 4a

A: Consider the given matrix A as,

Q: Let L be the line through the point Q=(-6, 9, -11) and direction vector [-1,2,-2]. Find the two…

A: Given Data: Let us consider the given L be the line through the point Q=-6, 9, -11 and direction…