Vector

What is Vector?

The vector is defined as the object that has both magnitude (size/length) and direction. Vectors are fundamental in mathematics, physics, and engineering for representing quantities that have both a numerical value and a specific orientation in space.

Key Concepts:

1. Magnitude: The length or size of the vector

2. Direction: The orientation the vector points toward

3. Representation: Typically shown as an arrow or written as coordinates

Examples: \(\vec{v} = (2, 3)\) OR \(\vec{a} = 2\vec{i} + 3\vec{j}\)

Common Examples:

1. Velocity: A car traveling 60 m/s northeast has both speed (magnitude) and direction

2. Force: Pushing an object with 10 Newtons of force toward the right

3. Displacement: Moving 5 meters north from your starting position

Scalar Product

The product of magnitudes of the two vectors with a cosine angle between them is called the Scalar Product. It is also known as the dot product.

Calculation of Scalar Product

1. Using magnitudes and the angle between vectors:

Let us suppose two vectors \(\vec{a}\) and \(\vec{b}\) and \(\theta\) being the angle between them. Then,

Note:

a. Condition of Perpendicularity: \(\mathbf{\vec{a}} \cdot \mathbf{\vec{b}} = 0 \)

b. Condition for being Parallel: \(\mathbf{\vec{a}} = n \mathbf{\vec{b}} \) where, \(n\) is a scalar.

2. Using vector components:

Let us suppose, \(\vec{a} = (a_1, b_1)\) and \(\vec{b} = (a_2, b_2)\). The dot product can then be calculated using the following formula.

Vector Geometry

Laws of Vector Addition

1. Triangle Law of Vector

The sum of vectors represented by two sides in an order is equal to the vector represented by the remaining side taken in the opposite order.

2. Parallelogram Law of Vector Addition

The sum of two co-initial vectors represented by the adjacent sides of a parallelogram is equal to the vector represented by the co-initial diagonal. This is called Parallelogram Law of Vector Addition. 

3. Polygon Law of Vector Addition

The polygon law of vector addition states that if vectors represented by the sides of a polygon taken in order, then the resultant vector is the vector represented by the closing side of the polygon taken in opposite order.

Some Important Formulae

1. Mid-Point Formula: \(\mathbf{\vec{m}} = \mathbf{\frac{\vec{a} + \vec{b}}{2}}\)

2. Section Formula

a. Internal Section Formula: \(\vec{p} = \frac{n\vec{a} + m\vec{b}}{m+n}\)

b. External Section Formula: \(\vec{p} = \frac{m\vec{b} - n\vec{a}}{m-n}\)

Different Theorems

Theorems Related to the Triangle

1. A line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it.

2. The line joining the vertex and the midpoint of the base of an isosceles triangle is perpendicular to the base.

3. The middle point of the hypotenuse of a right-angled triangle is equidistant from its vertices.

Theorems Related to Quadrilateral

1. The lines joining the middle points of the sides of a quadrilateral taken in order form a parallelogram.

2. The diagonals of a parallelogram bisect each other.

3. The diagonals of a rectangle are equal to each other.

4. The diagonals of a rhombus intersect at a right angle.

5. The angles at the circumference in a semi circle is a right angle.

Practice Questions Vector

Here are some Practice Questions for Chapter 6: Vector, Optional Math Class 10 (SEE)

1. Write the formula for finding the angle between \(\vec{a}\) and \(\vec{b}\).
2. If \(\vec{p} = 2\vec{i} + 3\vec{j}\), \(\vec{q} = -a\vec{i} + 4\vec{j}\) and \(\vec{p} \cdot \vec{q} = 0\), find the value of \(a\).
3. If \(\vec{a} = 4\vec{i} + 2\vec{j}\) and \(\vec{b} = -\vec{i} + 2\vec{j}\), find the angle between \(\vec{a}\) and \(\vec{b}\).
4. Find the position vector of mid-point of the line segment joining the points having position vectors \(5\vec{i} - 2\vec{j}\) and \(3\vec{i} + 6\vec{j}\).
5. The position vectors of athe vertices \(A\) and \(B\) are \(2\vec{i} + 3\vec{j}\) and \(\vec{i} - 3\vec{j}\) respectively. If point \(P\) divides \(AB\) in the ratio \(2:3\) then find the position vector of \(P\).
6. Prove by vector method that the line joining the vertex and the midpoint of the base of an isosceles triangle is perpendicular to the base.
7. MNOP is a rectangle. Prove by vector method that \(MO = NP\).