What is the magnitude of $\overrightarrow{A B}$ ?

Question

Prepp Team · Accepted Answer

Understanding Position Vectors and Vector Magnitude

The problem provides the position vectors of two points, A and B. A position vector of a point is a vector from the origin (0,0,0) to that point. Let $\vec{OA}$ be the position vector of point A, and $\vec{OB}$ be the position vector of point B.

Given:

Position vector of A, $\vec{OA} = \hat{i} - \hat{j}$
Position vector of B, $\vec{OB} = \hat{j} + \hat{k}$

We are asked to find the magnitude of the vector $\overrightarrow{AB}$. The vector $\overrightarrow{AB}$ represents the displacement from point A to point B. It can be calculated by subtracting the position vector of the initial point (A) from the position vector of the terminal point (B).

Calculating the Vector $\overrightarrow{AB}$

The formula to find the vector connecting two points A and B with position vectors $\vec{OA}$ and $\vec{OB}$ is:

$\overrightarrow{AB} = \vec{OB} - \vec{OA}$

Substituting the given position vectors:

$\overrightarrow{AB} = (\hat{j} + \hat{k}) - (\hat{i} - \hat{j})$

Now, we perform the subtraction by combining the coefficients of the respective unit vectors ($\hat{i}$, $\hat{j}$, $\hat{k}$):

$\overrightarrow{AB} = \hat{j} + \hat{k} - \hat{i} + \hat{j}$

Combine like terms:

$\overrightarrow{AB} = -\hat{i} + (1+1)\hat{j} + \hat{k}$

$\overrightarrow{AB} = -\hat{i} + 2\hat{j} + \hat{k}$

So, the vector $\overrightarrow{AB}$ is $-\hat{i} + 2\hat{j} + \hat{k}$.

Calculating the Magnitude of $\overrightarrow{AB}$

The magnitude of a vector $ \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} $ in three-dimensional space is given by the formula:

$ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} $

For the vector $\overrightarrow{AB} = -\hat{i} + 2\hat{j} + \hat{k}$, the components are $a = -1$, $b = 2$, and $c = 1$.

Now, we calculate the magnitude:

$ |\overrightarrow{AB}| = \sqrt{(-1)^2 + (2)^2 + (1)^2} $

$ |\overrightarrow{AB}| = \sqrt{1 + 4 + 1} $

$ |\overrightarrow{AB}| = \sqrt{6} $

Therefore, the magnitude of the vector $\overrightarrow{AB}$ is $\sqrt{6}$.

Summary of Steps

Identify the position vectors of points A and B.
Calculate the vector $\overrightarrow{AB}$ by subtracting $\vec{OA}$ from $\vec{OB}$.
Find the magnitude of the resulting vector $\overrightarrow{AB}$ using the formula $ \sqrt{a^2 + b^2 + c^2} $.

Following these steps, we found the magnitude of $\overrightarrow{AB}$ to be $\sqrt{6}$.

Revision Table: Vector Concepts

Concept	Description	Formula/Example
Position Vector	Vector from the origin (0,0,0) to a point P. Denoted as $\vec{OP}$.	If P is (x,y,z), $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$.
Vector Between Two Points (AB)	Vector from point A to point B. Calculated using position vectors.	$\overrightarrow{AB} = \vec{OB} - \vec{OA}$
Magnitude of a Vector	The length of the vector. For $ \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} $.	$ \|\vec{v}\| = \sqrt{a^2 + b^2 + c^2} $

Additional Information on Vector Operations

Vectors are fundamental quantities in physics and mathematics, possessing both magnitude and direction. Operations like addition, subtraction, and finding magnitude are crucial for solving problems involving forces, velocities, displacements, and more.

Vector Subtraction: Subtracting vector $\vec{A}$ from vector $\vec{B}$ ($\vec{B} - \vec{A}$) can be visualized as adding vector $\vec{B}$ and the negative of vector $\vec{A}$ (which has the same magnitude as $\vec{A}$ but points in the opposite direction). Component-wise, if $\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then $\vec{B} - \vec{A} = (b_1-a_1)\hat{i} + (b_2-a_2)\hat{j} + (b_3-a_3)\hat{k}$. This is exactly what we did when calculating $\overrightarrow{AB}$.

Magnitude: The magnitude of a vector represents its length or size. In a 3D Cartesian coordinate system, it's calculated using the Pythagorean theorem extended to three dimensions. It's always a non-negative scalar value.

Understanding position vectors allows us to represent points in space as vectors and use vector algebra to find distances (magnitudes of displacement vectors) and directions between points.