Chapter 11 Three Dimensional Geometry

Notes

Chapter 10: Vector Algebra – Complete Notes

1. Physical Quantities

A physical quantity is a quantity that can be measured. Physical quantities are mainly classified into two types: scalar quantities and vector quantities.

1.1 Scalar Quantities

A scalar quantity is a physical quantity that is completely described by its magnitude alone.

Examples: Time, length, distance, etc.

For example, if the distance between two places is 10 km, the value 10 km completely describes the distance. No direction is required.

1.2 Vector Quantities

A vector quantity is a physical quantity that has both magnitude and direction.

Examples: Force, acceleration, momentum, etc.

A vector is generally represented by a lowercase or uppercase letter with an arrow above it, such as \(\vec{a}\) or \(\vec{A}\).

If A and B are two points, then the vector from A to B is represented by \(\overrightarrow{AB}\).

When the direction of a vector is reversed, its sign changes.

\[\overrightarrow{AB}=-\overrightarrow{BA}\]

2. Representation of a Vector in Three Dimensions

Consider a point P(a, b, c) in three-dimensional space. The vector joining the origin O to the point P can be written as:

\[\vec{r}=a\hat{i}+b\hat{j}+c\hat{k}\]

Here, \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) are unit vectors along the positive x-axis, y-axis, and z-axis, respectively.

3. Magnitude of a Vector

The magnitude of a vector represents its length.

If

\[\vec{a}=a\hat{i}+b\hat{j}+c\hat{k}\]

then its magnitude is:

\[|\vec{a}|=\sqrt{a^2+b^2+c^2}\]

The magnitude of a vector is always non-negative.

4. Unit Vector

A unit vector is a vector whose magnitude is equal to 1.

The unit vector in the direction of a non-zero vector \(\vec{a}\) is:

\[\boxed{\hat{a}=\frac{\vec{a}}{|\vec{a}|}}\]

Also,

\[|\hat{a}|=1\]

For example, suppose a vector is given by \(2\hat{i}+3\hat{j}+\hat{k}\). To find its unit vector, first calculate its magnitude and then divide the vector by that magnitude.

Solved Example 1: Find the Unit Vector