Please answer and explain questions above. Given the vectors P = 3i + 4j and Q =

Question

Please answer and explain questions above.

Given the vectors P = 3i + 4j and Q = 2i + 2k, find a vector of unit length perpendicular to both P and Q. Write down the results of all possible cross products between i, j, and k. Prove that A times B = -B times A for any two vectors A and B. Prove that A times A = 0 for any vector A. Prove that the determinant form of the cross product renders the same result as the component form. (a). Draw the forces acting on the ladder shown below if there is no friction on the vertical surface. (b) What is the torque equation for an axis located at the bottom of the ladder? (c) What is the torque equation for an axis located at the top of the ladder? (d) What is the torque equation for an axis located at the point in space labeled P? Draw the forces acting on the rod shown below and write the torque equation for the left end. A uniform sheet of metal in the shape of a hemisphere of radius 1 m is used as a sign. The hemisphere has a mass of 200 kg and is suspended by two vertical wires attached to points along its horizontal diameter. One wire is attached to the left end of the diameter and the other is attached to a point three-quarters of the way to the other end. (a) Write the torque equation for the left end of the diameter. (b) Write the torque equation for the right end of the diameter.

Explanation / Answer

Suppose vector R is perpendicular to both P and Q, then

P.R = 0 And Q.R = 0

P.R = (3i + 4j).(ai + bj + ck)

P.R = 3*a + 4*b

3a + 4b = 0

b = -3a/4

Q.R = (2i + 2k).(ai + bj + ck)

Q.R = 2a + 2c = 0

2a + 2c = 0

c = -a

Vector R is of unit length which means

|R| = sqrt (a^2 + b^2 + c^2) = 1

squaring both side

(a^2 + b^2 + c^2) = 1

(a^2 + (-3a/4)^2 + (-a)^2) = 1

2a^2 + 9a^2/16 = 1

41a^2/16 = 1

a = sqrt (16/41) = 4/sqrt 41

c = -a = -4/sqrt 41

b = -3a/4 = -3/sqrt 41

Vector R = (1/sqrt 41)*(4i - 3j - 4k)

4.

ixj = k

jxi = -k

jxk = i

kxj = -i

kxi = j

ixk = -j

ixi = 0

jxj = 0

kxk = 0

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