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Tutorial 2


1. Let u = (3; 1;􀀀1; 5) and v = (0; 2; 1;􀀀3). Find scalars a and b so that au + bv = (3;􀀀3;􀀀3; 11).

2. Find the initial point of a non-zero vector u with terminal point Q(3; 0;􀀀5)such that u is oppositely directed to v = (4;􀀀2;􀀀1).

3. For u = (􀀀2;􀀀1; 4; 5), v = (3; 1;􀀀5; 7) and w = (􀀀6; 2; 1; 1), evaluate k3u 􀀀 5v + wk and k3uk 􀀀 k5vk + kwk.
4. Find the Euclidean distance and the cosine of the angle between the vec-tors u = (3; 3; 3) and v = (1; 0; 4).

5. Given vector u = (3;􀀀4), nd a unit vector that (i) has the same direction as u, (ii) is oppositely directed to u.

6. Determine if the points A(1; 1; 1);B(􀀀2; 0; 3) and C(􀀀3;􀀀1; 1) form the vertices of a right angle triangle.

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7. Find the work done by the force F = (3;􀀀2; 5) in moving a particle from a point P(1; 4;􀀀1) to a point Q(􀀀2; 3;


1). Use the relation Work = F  r,where r is the displacement.

8. Find the point-normal form of the equation of a plane passing through P(􀀀1; 3;􀀀2) and having normal n = (􀀀2; 1;􀀀1).

9. Determine whether the planes 4x 􀀀 y + 2z = 5 and 7x 􀀀 3y + 4z = 8 are parallel.

10. Given vectors u = (6; 2) and a = (3;􀀀9), determine the vector component of u along a and the vector component of u orthogonal to a.

11. Find the vector and parametric equations of a plane containing the point(-3,1,0) and the parallel vectors v1 = (0;􀀀3; 6) and v2 = (􀀀5; 1; 2).

12. The position vectors of the points A and B are (1; 4; 6) and (3; 4; 7), respectively. Find the vector equation of the line AB and nd the points where the line intersects the coordinate planes.
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