Question

Find (a) north cross west (b) down dot south (c) east cross up (d) west dot west and (e) south cross south. Let each vector have unit magnitude.

Answer 1

Hint: We know that the dot product of two unit vectors is zero when they are perpendicular to each other and it is one if they are parallel to each other. Also, the cross product of two unit vectors is zero when they are parallel to each other. If the two unit vectors are perpendicular to each other, the resultant is another unit vector perpendicular to the both unit vectors. Assign the unit vector for each direction.

Complete step by step answer:
The dot product of unit vectors \[\hat i\] and \[\hat j\] is represented as, \[\hat i \cdot \hat j\] while the cross product of two vectors is represented as \[\hat i \times \hat j\].

We know that the dot product of two unit vectors is zero when they are perpendicular to each other and it is one if they are parallel to each other. Also, the cross product of two unit vectors is zero when they are parallel to each other. If the two unit vectors are perpendicular to each other, the resultant is another unit vector perpendicular to the both unit vectors.
\[\hat i \times \hat j = \hat k\]
\[\hat k \times \hat i = \hat j\]
\[\hat j \times \hat k = \hat i\]

We assume \[\hat i\] points towards East, \[\hat j\] points towards north and \[\hat k\] points towards upward.

North cross west
\[\hat j \times \left( { - \hat i} \right) = \hat i \times \hat j = \hat k\]
Therefore, the north cross west is in an upward direction.

Down dot south
\[ - \hat k \cdot \left( { - \hat j} \right) = 0\]
Therefore, down dot south is zero.

East cross up
\[\hat i \times \hat k = - \hat j\]
Therefore, the east cross up is southward.

West dot west
\[\left( { - \hat i} \right) \cdot \left( { - \hat i} \right) = 1\]
Therefore, west dot west is 1.

South cross south
\[\left( { - \hat j} \right) \times \left( { - \hat j} \right) = 0\]
Therefore, south cross south is zero.

Note:
Students should remember the cross products of two unit vectors. We know that the cross product \[\hat i \times \hat j = \hat k\]. But if the cross product is \[\hat j \times \hat i\], then the resultant is \[ - \hat k\]. In this way recall every cross product.