(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2),
b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to
(2) Suppose a, b, c are non zero vectors.
a. Explain why (a x b) x (a x c)
Two vectors having equal magnitudes (A) makes an angle z with each other. Find the magnitude anddirection of the resultant and prove that the resultant of two equal vectors bisects the angle between them.
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Suppose that a mountain has the shape of an elliptic paraboloid , where a and c are constants, x and y are the east-west and north-south map coordinates and z is the altitude above the sea level (x,y,z are measured all in metres). At the point (1,1), in what direc
Suppose A(3,-1,0) and B(-4,-2,3) are 2 points in 3-space. Find a vector with the following three characteristics: initial point at the origin, collinear but in the opposite direction of vector AB , length 3
1) A particle is moving in R^3 so that at time t its position is r(t) = (6t, t^2,t^3).
a. Find the equation of the tangent line to the particle's trajectory at the point r(1).
b. The particle flies off on tangent at t0 = 2 and moves along the tangent line to its trajectory with the same velocity that it had at time 2. (Note:
1) V(x, y, z) = (x + y + z)2 i + (x + y)2 j + x2 k. Find div V(3, 2, 4) ≡ ∇? V (3, 2, 4)
2) F (x, y) = xe2y i + y/(x + y) j. Find ∇ ? F (4, 0)
3) F (x, y, z) = -yz i + xz j - xy k. Find curl F (1, 2, 5) = ∇×F ( 1, 2, 5)
4) Line L is defined by the following equation:
Line M is defined by:
Find a and b, if known that lines are parallel to each other.
5) Line L is defined by intersection of two planes
2x+3y?z = 1 and -2x+y+2z = 0
Plane P is defined by -2x + y ?z = 4
Find any directional vector for the line, point of intersection of line a
(1) A force F of magnitude 6 in the direction i - 2j + 2k acts at the point P = (1,-1, 2).
a. Find the vector moment M of F about the origin.
b. Find the components of M in the direction of the (positive) x - axis, y -axis and z -axis.
c. Find the component of M about an axis in the direction