Tutorial 1 Solution Emagnet

TUTORIAL 1

(SOLUTION)

EEE 3133 ELECTROMAGNETIC FIELDS AND WAVES DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING 1.

Find the unit vector along the line joining point (2,4,4) to point (-3,2,2) (Ans:-0.870ax -0.348ay-0.348az )

2.

Let A = 2ax+5ay-3az, B = 3ax-4ay, and C = ax+ay+az. Calculate: (a) A + 2B (b) |A-5C| (c) the values of k if |kB| = 2 (d) (AxB) / (A.B) (Ans: 8ax -3ay-3az, 8.544, ±0.4, 0.857ax + 0.643ay+ 1.643az )

3.

Given:

A = 2ax + ay - 3az B = ay – az C = 3ax + 5ay + 7az

Determine:

(a) A – 2B + C (b) C – 4(A + B) (c) 2A – 3B |C| (d) A·C - |B|² (e) ½B x (1/3A + 1/4C)

(Ans: 5ax +4ay+6az, -5ax -3ay+23az, 0.439ax - 0.11ay -0.329az,-12, 1.167ax - 0.708ay -0.708az )

4.

If the position vectors of points T and S are 3ax-2ay+az and 4ax+6ay+2az, respectively. Find: (a) coordinates of T and S (b) distance vector from T to S (c) distance between T and S (Ans:T(3,-2,1), S(4,6,2), ax +8ay+az, 8.124)

5.

A = 5ax + 3ay + 2az B = -ax+4ay +6az C = 3ax + 2ay Find values of α and β such that αA+βB+C is parallel to the y-axis Given:

(Ans:α = -12/7,β = -4/7)

6.

A = αax + ay + 4az B = 3ax + βay – 6az C = 5ax - 2ay + γaz Calculate α, β, γ such that the vectors are mutually orthogonal If

(Ans:α =2.667,β =16,γ = -2.833)

7.

Given:

P = -2ax - ay - 2az Q = 4ax + 3ay + 2az R = -ax + ay + 2az

Find:

(a) |P + Q - R| (b) P · Q x R (c) Q x P ·R (d) (P x Q) · (Q x R) (e) (P x Q) x (Q x R) (f) cos θPR (g) sin θPQ (Ans:3.742,-12,12,42, -48ax -36ay-24az, 114.1°, 21.8° )

8.

If A = -ax+6ay+5az and B=ax+2ay+3az, determine: (a) the scalar projections of A on B (b) the vector projection of B on A (c) unit vector perpendicular to the plane containing A and B (Ans:6.949, -0.419ax +2.516ay+2.097az, 0.577ax +0.577ay -0.577az)

9.

Find the angle that vector H = 3ax + 5ay – 8az makes with x, y, and z-axes. (Ans:α=72.36°,β =59.66°,γ=143.91° )

10.

P = 2ax - ay - 2az Q = 4ax + 3ay + 2az R = -ax + ay + 2az Calculate triple scalar product of P,Q, and R (Ans:5)

11.

The vertices of a triangle are located at (4,1,-3), (-2,5,4), and (0,1,6). Find the three angles of the triangle. (Ans:28.48°,73.47°,78.04°)

12.

Points P,Q, and R are located at (-1,4,8), (2,-1,3), and (-1,2,3) respectively. Determine: (a) distance between P and Q (b) distance vector from P to R (c) angle between QP and QR (d) area of triangle PQR (e) perimeter of triangle PQR (Ans:7.681, -2ay -5az,137.43°, 11.02,17.31)

13.

E and F are vector fields given by E = 2xax + ay + yzaz and F = xyax - y²ay + xyzaz. Determine: (a) |E| at (1,2,3) (b) the component of E along F at (1,2,3) (c) a vector perpendicular to both E and F at (0,1,-3) whose magnitude is unity (Ans: 6.403,1.286ax -2.571ay+3.857az, ± ax )