Question

What is the terminal point of a vector?

Answer 1

Hint: First we need to understand the term vector how it is represented. For understanding the representation of vector we will consider two points in a plane as $$A(x_1,y_1,z_1)$$ and $$B(x_2,y_2,z_2)$$ and the process to find its direction. Now, we will understand the initial and terminal point of a vector using some examples.

Complete step-by-step solution:
Here we have been asked to define the term terminal point of a vector. First let us understand the definition of a vector quantity and how we represent a vector.
Now, in mathematics a vector quantity is a quantity that has both magnitude and direction. For example: - weight, force, displacement, velocity, acceleration etc. If we mention a vector only with its magnitude then it will not make any sense. For example: - if we say that a particle has a velocity of 5 m/s then it is of no meaning but we need to specify the direction like the velocity of a particle is 5 m/s along the north – east direction. If we will consider only the magnitude then it will be its speed which is a scalar quantity.
A vector is represented by an arrow $(\to )$ where the head of the arrow represents the direction of the vector. For example: - let us consider two points in a plane given as $$A(x_1,y_1,z_1)$$ and $$B(x_2,y_2,z_2)$$ then the vector from A to B will be denoted as $\overrightarrow{AB}$ and is given as:
$\Rightarrow \overrightarrow{AB}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}............\left(i\right)$
Here, $\hat{i},\hat{j},\hat{k}$ represents the unit vectors along the positive x, y, z direction respectively. They are called unit vectors because their magnitude is 1 according to the convention. Now, here the point $$A(x_1,y_1,z_1)$$ is called the initial point because the vector is originating from here and point $$B(x_2,y_2,z_2)$$ is called the terminal point because the vector is ending here.

Let us consider an example where the initial point is A (2, 3, 5) and the vector AB is given is $$\overrightarrow{AB}=-\hat{i}+10\hat{j}+4\hat{k}$$ and we have to determine the terminal point.
Now, let us assume the terminal point as B(x, y, z). Therefore, using relation (i) we can write:

$\Rightarrow -\hat{i}+10\hat{j}+4\hat{k}=(x-2)\hat{i}+(y-3)\hat{j}+(z-5)\hat{k}$
On comparing the coefficients of $\hat{i},\hat{j},\hat{k}$ both the sides we get,
$\Rightarrow x-2=-1$, $y-3=10$ and $z-5=4$
$\Rightarrow x=1$, $y=13$ and $z=9$
Hence, the terminal point of the vector AB is B (1, 13, 9).

Note: Note that if you are provided with the vector and its terminal point then you can use relation (i) to get the answer and similarly if terminal and initial points are given then we can write the required vector. If we have to reverse the direction of a vector then we just multiply it with -1. The above concepts are very helpful in physics while studying about the position vectors.