MTE1_A_Solution

# MTE1_A_Solution - 1 . as p 3.5rad q 4.8rad a Definez=p q. z...

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EEL 3105 Mid Term Exam 1A Solution 1. Consider two vectors p and q in two dimensions. Suppose p and q are given in polar coordinates as 3. 5 4. 8 pr a d qr a d  a. Define z = p+q. Suppose z is expressed as j zr e . Find r and . [2pts] b. Find the scalar product of p and q. [2pts] c. Find the cross product of p and q. [2pts] Answer: First, we convert p and q to Cartesian coordinates.    3cos(0.5) 3sin(0.5) [2.63 1.44] 4cos(0.8) 4sin(0.8) 2.79 2.87 p q  a. z=p+q=   5.42 4.31 . We need to express z in polar coordinates to get r and . We can calculate r and to get 22 5.42 4.31 6.92 4.31 arctan 0.67 5.42 r rads     b. Scalar product is easy to calculate since we already have the Cartesian representation of p and q. Thus . 2.69*2.79 1.44*2.87 11.47 pq  Another way to get this answer is to use the polar representation. Angle between p and q is [0.8 0.5]rads. Therefore, the scalar product is 12cos(0.3)=11.47. c. Cross product of p and q will be a vector perpendicular to the plane of p and q given by the right hand rule. Let’s represent p and q in the x,y plane via unit vectors along x and y directions, i, j. Thus, 2.63 1.44 2.79 2.87 2.63*2.87 1.44*2.79 3.54 pij qij k k  where k is the unit vector along the z direction. Another way to get this answer is from the polar representation for p and q. Again, angle between p and q is 0.3rads.

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Therefore, the magnitude of pxq is 12sin(0.3)=3.54 and its direction is perpendicular to the plane of p and q. 2. Suppose p and q are two vectors in 3 dimensions. Magnitude of p is 3 and the magnitude of q is 5. Moreover, it is known that the magnitude of the scalar product of p and q is equal to the magnitude of the cross product of p and q. a. Find the value(s) of the scalar product of p and q. [2pts] b. Suppose we change p and q so that the magnitudes of p and q remain 3 and 5 respectively but the angle between p and q is doubled. What is the scalar product of p and q? [2pt] c. Suppose p and q are changed as described in part b above. What is the magnitude of the
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MTE1_A_Solution - 1 . as p 3.5rad q 4.8rad a Definez=p q. z...

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