Question

Let vector $\overrightarrow{\mathbf{A}}$point from the origin into the second quadrant of the $\mathit{x}\mathit{y}$plane and vector $\overrightarrow{\mathbf{B}}$point from the origin into the fourth quadrant. The vector$\overrightarrow{\mathbf{B}}\mathbf{-}\overrightarrow{\mathbf{A}}$ must be in which quadrant,

(a) the first,

(b) the second,

(c) the third, or

(d) the fourth, or

(e) is more than one answer possible?

Short answer

The correct option is (d) .

Step-by-step solution

 Step 1: Define the vector quantity

A quantity that has both magnitude and direction is called a vector quantity. It's usually represented by an arrow with the same direction as the amount and a length proportionate to the magnitude of the quantity.

Determine the position of the vector.

The $\overrightarrow{B}-\overrightarrow{A}$ vector can be written as follows;

$\overrightarrow{B}-\overrightarrow{A}=\overrightarrow{B}+\left(-\overrightarrow{A}\right)$

Here,$-\overrightarrow{A}$is the vector that has the same magnitude as the vector$\overrightarrow{A}$,but the opposite direction to vector$\overrightarrow{A}$.

The position of the vectors is shown in the diagram below.

Here, $\overrightarrow{A}$ and $\overrightarrow{B}$ are two random vectors.

The second quadrant contains the vector $\overrightarrow{\mathbf{A}}$, while the fourth quadrant contains the vector$\overrightarrow{B}$.

The vectors$-\overrightarrow{A}$and $\overrightarrow{B}$ are in the fourth quadrant in the diagram above.

The resultant of the vectors$\overrightarrow{B}$and$-\overrightarrow{A}$is the vector$\overrightarrow{B}-\overrightarrow{A}$.

As a result, the vectorrole="math" localid="1663738658878" $\overrightarrow{B}-\overrightarrow{A}$ is in the fourth quadrant. Hence, options (a), (b), (c), and (e) are incorrect.

As a result, the correct choice is

Explanation for option (a)

The vector $\overrightarrow{B}-\overrightarrow{A}$ is in the fourth quadrant .Thus the option (a) is incorrect.

Explanation for option (b)

The vector is in the fourth quadrant .Thus the option (b) is incorrect.

Explanation for option (c)

The vector is in the fourth quadrant .Thus the option (c) is incorrect.

Explanation for option (e)

The vector is in the fourth quadrant .Thus the option (e) is incorrect.