18 - University Physics with Modern Physics

Question

If $\overrightarrow{\mathit{A}}$ and $\overrightarrow{\mathbf{B}}$ are nonzero vectors, is it possible for both $\overrightarrow{\mathbf{A}}\mathbf{·}\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{A}}\times \overrightarrow{\mathbf{B}}$ to be zero? Explain.

 

Step-by-Step Solution

Answer

Yes, it is possible both to have $\overrightarrow{A}·\overrightarrow{B}$  and $\overrightarrow{A}\times \overrightarrow{B}$ zero value.

Vector Product

There are the two types of vector multiplication, dot product and cross product. The result of dot product is scalar whereas the result of cross product is vector.

Explanation

The expression for the dot product and cross production is shown below,

$$\begin{gathered} \overrightarrow{A}·\overrightarrow{B}=\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \theta \\ \overrightarrow{A}\times \overrightarrow{B}=\left|\overrightarrow{A}\right|\sin \theta \hat{n} \end{gathered}$$

Here, θ is the angle between the two vectors.

 

From above equation, it is clear that when the value of θ for dot product is 90 degree and for cross product is 0 degree, then the value of $\overrightarrow{A}·\overrightarrow{B}$and $\overrightarrow{A}·\overrightarrow{B}$ will be zero.