(d) Let r, m and n be the direction cosines of the line L. Show that  r2+ m2+n2=1

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A: Given vectors:A= <8t, 2> and  B= <4, -1>

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Q: For what value of t, if any, is the vector (4t, -1) parallel to the vector (4, -1) ?

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Q: If a = 2i -j+2k and b =i+2j-2k, find two vectors c and d satisfying all the following conditions:…

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Q: What angle does the line of intersection of the planes (x 2y + 3z 5 and 4x+y-2z-7 make with the…

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Q: 1. Give the vector equation of a line perpendicular to the line L: x= 21-1, y=-31, 1 e R. Are there…

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Q: For what value of t , if any, is the vector (4t, -1) parallel to the vector (4, -1) ?

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Q: 16. The accompanying figure shows nonzero vectors v, w, and z, with z orthogonal to the line L, and…

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Q: The vectors T and N, when defined, (a) are parallel. (b) are orthogonal. (c) have a zero cross…

A: T is the tangent vector and N is the normal vector.  So they are always orthogonal to each other.

Q: find a. the direction of vector P1P2 and b. the midpointof line segment P1 P2. If vector AB = i + 4j…

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Q: 1. The two unit vectors parallel to the line y = x- 3 are and

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Q: 2. Let each of the vector A = -8a, + 2ay – 4a,, B = 3ax – 4ay |3D Z> + 5a,, and C = 4ax + 5a, extend…

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Q: 2. If å and b are two vectors with lengths 2 and 5 and angles between 150°, respectively, obtain the…

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Q: Owing vectors is perpendicular to the line L: 2x +3y- 6 = 0 2i + 3j O 2i - 3j (후 +1를 이 rious

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Q: The unit vector which is perpendicular to both r1=8ax+3ay-2az, * r2=2ax+7ay-6az is equal to a)…

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Q: If a, b, and c are constant vectors, r is the position vector zi+ yj+ zk, and Eis given by the…

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Q: x – 1 1- x z – 3 Consider the lines l1 : = 2 – y = z + 1 and l2 : = y – 2 = 3 1. Determine a…

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Q: For two nonzero vectors u and v, if u v=0 then u and v are parallel.

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Q: 2. Given two non-parallel lines l1: r = 2+t, y = 4+t, z = 8, and l2: x = 3, y = 2 + t, z = 5+t, (a)…

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Q: A = x6- y9+z3 B = x6- y3+z9 C = x 2- yl-zl For the given vectors A,B, and C show that whether C is…

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Q: The lines / and m have equations r 3i- 2j+ k+1(-i+2j + k) and r 4i + 4j +2k + u(ai+ bj- k)…

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Q: 1. The two unit vectors parallel to the line y 5x + 2 are and 2. The two unit vectors parallel to…

A: 1. Put x=t, then y=5t+2 and the parametric equation of the line is rt=t,5t+2=0,2+t1,5 and parallel…

Q: a) Find the distance between two skew lines L1 and L2 such that L1: 1+ 4t, y=5-4t, z=-1+5t  L2:…

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Q: Consider a point Q(x1, y1, z1) and a line L given by the vector equation r = r0 +tv,where r0 =…

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Q: State the condition that the two vectors i and y are perpendicular. Find the value(s) of k such that…

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Q: 1. If a and b are two non-parallel vectors and the position vectors of P, Q and R are 3. +-b and Aa…

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Q: Find a vector equation of the line r through D4 and perpendicular to the line s through F4 and B1.…

A: Given,D4→ = 1i ^ + 2j^ + 3k^Equation of line passing through D4→ is:x-1a= y-2b=z-3cThe above line is…

Q: Find the values of x, y and z that satisfy the vector equation 1 3 ·¤-A-D-A-A-D 3 +y-2 +2 1 1-2 +3…

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Q: For what value of t , if any, is the vector (2, -4t) parallel to the vector (4, -1) No t O A. 1 %3D…

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Q: 1. Show that the vectors u̟ and uz parallel to lines L1 and L2 respectively, are orthogonal. L1: x =…

A: The given equations of lines are: L1:x=5-2t, y=-7+t, z=6+4t and L2:x=2+3t, y=5+2t, z=1+t Rewriting…

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Q: Consider the lines 3 - z x – 5 l1 : 2 and I. Find two unit vectors that are parallel to l1.

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Q: 14. Given that a and bare two vectors, then the vector (a + b) × (a – b) is A. perpendicular to (a –…

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