1. Two planes P and P, are defined by the equations x+y+z=1 and 2x-y+z=0, respectively. The planes intersect along a line L. Write down two vectors which are normal to P, and P, respectively. (a) (b) (c) (d) Hence find a vector running in the direction of the line L. Find the point lying in both planes P, and P, which has z = 0. Hence obtain the equation of the line L.

For what value of a and b are the vectors (a-b,2) and (4,a+b) equal?

Given the three nonzero vectors A, B, and C, show that if A ⋅ (B x C) = 0, the three vectors must lie in the same plane.

Determine the rectangular representations of the vector F.

curl of the vector field 2x3yi

II. Let f(x, y) = e*-y. (a) Evaluate f(2, 4). (b) Find the domain of f. (c) Find the range of f.

Q4) Let vector A = 2i+3j , vector B %3D 2i-2j, and vector C = (-A) - (2B). (a) %3D Write vector C in component form. (b) Draw a coordinate system and on it show the three vectors (A, B and C). (c) what are the magnitude and * ?direction of vector C

Prove that for any vector u, u = (u e, )e, + (ue, )e, +(u e,)e,. [Hint: write u in component form.] Find the projection of the vector u = e, - 2e, +e, on the vector 2. 3. v = 4e, - 4e, +7e,.

- If the point C (2,4,-1) is the midpoint between the points (a,b +1,c +2), (a,b+3,c), find the values of a,b and c.

1) Sketch the vectors in a,b,c, and d using the provided figures A) a. u + v b. u + v + w C.u -v d. u- w B) a. u - v b. u - v + w c. 2u - v d. u + v + w 2) A) Linear combination Let u = 2i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av + bw. Linear combination Let u =i - 2j, v = 2i + 3j, and w = i + j. Write u = u, + u, where u, is parallel to v and u, is parallel to w. (See Exercise 41.) B) Velocity An airplane is flying in the direction 25° west of north at 800 km/h. Find the component form of the velocity of the air- plane, assuming that the positive x-axis represents due east and the positive y-axis represents due north. 3) Rectangle Find the measures of the angles between the diago nals of the rectangle whose vertices are A = (1, 0), B = (0, 3), C = (3, 4), and D = (4, 1). 4) Triangle Find the measures of the angles of the triangle whose vertices are A = (-1, 0), B = (2, 1), and C = (1, –2).