Question

Which of the following are vectors?

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{}\overrightarrow{\mathbf{r}}\mathbf{/}\mathbf{2} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{\left|}\overrightarrow{\mathbf{r}}\mathbf{\right|}\mathbf{/}\mathbf{2} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}<r_x,r_y,r_z> \\ \mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{5}\mathbf{·}\overrightarrow{\mathbf{r}} \end{gathered}$$

Short answer

(a) $\overrightarrow{r}/2$is vector quantity

(b)$\left|\overrightarrow{r}\right|/2$ is a scalar quantity

(c) The given mathematical entity is representing the position vector r so it is a vector quantity.

(d)$5·\overrightarrow{r}$ is vector quantity

Step-by-step solution

Vectors

A vector is directional position representation of any entity. A vector can be shown by an arrow on the top of the mathematical values.

Vectors are those mathematical entities who shows their own mathematical operations with other vector or scalar entities.

• Multiply or divide a vector by a scalar it will be vector
• The mod of a vector will be scalar
• Adding one vector to another vector, the resultant will be vector
• Subtract one vector to another vector, the resultant will be vector
• Dot product of two vectors will be scalar
• Cross product of two vectors will be vector
• Differentiation of a vector will be vector

Step 2: (a)r→/2

According to the vector operations the multiply or divide a vector by a scalar it will be vector, the given mathematical entity vector r is getting divide by the scalar quantity 2 the resultant will be vector. Hence$\overrightarrow{r}/2$ is vector quantity.

Step 3: (b)|r→|/2

According to the vector operations the the mod of a vector will be scalar. In the given mathematical entity the mod of vector r is the scalar quantity and diving it by 2 will give the resultant a scaler. Hence $\left|\overrightarrow{r}\right|/2$is a scalar quantity.

Step 4: (c)<rx,ry,rz>

The given mathematical entity is representing the position vector r so it is a vector quantity.

Step 5:  (d)5·r→

According to the vector operations the multiply or divide a vector by a scalar it will be vector, the given mathematical entity vector r is getting multiply by the scalar quantity 5 the resultant will be vector. Hence$5·\overrightarrow{r}$ is vector quantity.