- Vector quantities have a magnitude as well as direction.
- For example, velocity is a vector quantity as it includes speed (magnitude) and direction.
Representation of a Vector
1. Component Form of Vector
- If a vector in XY-plane has initial point and terminal point , its component form is expressed as
- The magnitude of is
2. Vector representation in terms of and
- If is a vector and , then .
Scalar Multiplication: The product of a vector or with a scalar is expressed as
- A vector whose magnitude is one is called a unit vector.
- In other terms, any non-zero vector divided by its magnitude is a unit vector.
- Mathematically, the unit vector of a vector is expressed as
- A unit vector is in the direction of .
Note: The unit vector parallel to the x-axis is , while the unit vector parallel to the y-axis is .
Horizontal & Vertical Components of a Vector
- If is the angle between the positive x-axis and a vector , then
- The horizontal component of v is .
- The vertical component of v is .
- The vector is represented as:
Dot Product of Vectors
- The dot product of two vectors , and is given by:
- If is the smallest non-negative angle between , and , then the dot product is given by:
Note: The dot product of two vectors is a real number, not a vector and hence it is also known as a scalar product.
Angle Between Two Vectors
- If is the smallest non-negative angle between and , then
Note: Two non-zero vectors u and v are perpendicular if and only if
Question 1: Given and , find .
Question 2: Find , find .
Question 3: Find a unit vector in the direction .
Question 4: Given and , find .
Question 5: Find the dot product of and .
Question 6: Find the measure of the smallest positive angle between the vectors and .
- If a vector in XY-plane has an initial point and the terminal point , its component form is: .
- The magnitude of is: .
- The component form of a vector with a direction angle is .
- A unit vector has a magnitude of 1.
- Any non-zero vector divided by its magnitude is a unit vector. If is a non-zero vector, then it's a unit vector .
- The sum of two vectors and is expressed as .
- The product of a scalar with a vector is .
- The direction angle of a vector is given by: .
- The angle between two vectors and can be found by the formula: where
- The unit vector parallel to the x-axis is , while the unit vector parallel to the y-axis is .
- The dot product of two vectors is a real number, not a vector; hence, it is also known as a scalar product.
- Two non-zero vectors, u and v, are perpendicular if and only if
- Abhishek Tiwari
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