Question

Let $\hat{\mathbf{i}}$be directed to the east, $\hat{\mathbf{j}}$be directed to the north, and be directed upward.What are the values of products (a) $\hat{\mathbf{i}}\mathbf{.}\hat{\mathbf{k}}\mathbf{}\mathbf{,}\left(b\right)\left(-\hat{k}\right)\mathbf{.}\left(\hat{j}\right)$, and (c)$\left(\hat{j}\right)\mathbf{.}\left(-\hat{j}\right)$? What are the directions (such as east or down) of products (d)$\hat{\mathbf{k}}\mathbf{\times }\hat{\mathbf{j}}\mathbf{}\mathbf{,}\mathbf{}\left(e\right)\left(-\hat{i}\right)\mathbf{\times }\left(-\hat{j}\right)\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\left(f\right)\left(-\hat{k}\right)\mathbf{\times }\left(-\hat{j}\right)$?

Short answer

$$\begin{gathered} \hat{\mathrm{i}}.\hat{\mathrm{k}}=0 \\ \left(-\hat{\mathrm{k}}\right).\left(-\hat{\mathrm{j}}\right)=0 \\ \left(\hat{\mathrm{j}}\right).\left(-\hat{\mathrm{j}}\right)=-1 \\ \hat{\mathrm{k}}\times \hat{\mathrm{j}}=-\hat{\mathrm{i}} \\ \left(-\hat{\mathrm{i}}\right)\times \left(-\hat{\mathrm{j}}\right)=\hat{\mathrm{k}} \\ \left(-\hat{\mathrm{k}}\right)\times \left(-\hat{\mathrm{j}}\right)=-\hat{\mathrm{i}} \end{gathered}$$

Step-by-step solution

To understand the concept

This problem is based on the product rule in which the vector product and scalar product are the two ways of multiplying vectors. In right handed coordinate system, thumb points towards the z axis. And curl of the fingers represents a motion from x axis to y axis. Here I directed to the east means x axis, j directed to the north means y axis and k directed upward means these are mutually perpendicular to each other.

Step 2:To find i^.k^

Since vector $\hat{\mathrm{i}}$and vector $\hat{\mathrm{k}}$are orthogonal

$\hat{\mathrm{i}}.\hat{\mathrm{k}}=0$

Step 3:To find (-k^).(-j^)

$\left(-\hat{\mathrm{k}}\right).\left(-\hat{\mathrm{j}}\right)=0$

Step 4:To find (j^).(-j^)

$\left(\hat{\mathrm{j}}\right).\left(-\hat{\mathrm{j}}\right)=-1$

Step 5:To findk^×j^

$\hat{\mathrm{k}}\times \hat{\mathrm{j}}=-\hat{\mathrm{i}}$

Step 6:To find(-i^)×(-j^)

$\left(-\hat{\mathrm{i}}\right)\times \left(-\hat{\mathrm{j}}\right)=\hat{\mathrm{k}}$

Step 7:To find(-k^)×(-j^)

$\left(-\hat{\mathrm{k}}\right)\times \left(-\hat{\mathrm{j}}\right)=-\hat{\mathrm{i}}$