CBSE Class 11-Mathematics: Addition of Vector Algebra in Short (For CBSE, ICSE, IAS, NET, NRA 2022)

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A vector is a quantity that requires both a magnitude and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as

Two vectors A and B can be added together to give another resultant vector C.

Subtraction of Vectors

Two vectors A and B can be subtracted to give another resultant vector D.

Scalar Multiplication of Vector

When we multiply any vector A with any scalar quantity then its direction remains unchanged and magnitude gets multiplied by . Thus,

Scalar multiplication of vectors is distributive i.e.. ,

Dot Product of Vectors

Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both the vectors are placed tail to tail.

Dot product is represented as thus,

where is the angle between two vectors?

Result of dot product of two vectors is a scalar quantity.

Dot product is commutative:

Dot product is distributive:

Cross Product of Two Vectors

Cross product or vector product of two vectors A and B is defined as

where is the unit vector pointing in the direction perpendicular to the plane of both A and B?

Result of vector product is also a vector quantity.

Cross product is distributive i.e.. , A x (B + C) = (A x B) + (A x C) but not commutative and the

cross product of two parallel vectors is zero.

In component form addition of two vectors is

where,

Thus, in component form resultant vector becomes,

Subtraction of Two Vectors

In component form subtraction of two vectors is

where,

Thus, in component form resultant vector D becomes,

NOTE:- Two vectors add or subtract like components.

Dot Product of Two Vectors

Thus, for calculating the dot product of two vectors, first multiply like components, and then add.

Cross Product of Two Vectors

Cross product of two vectors is itself a vector.

To calculate the cross product, form the determinant whose first row is , whose second row is A (in component form) , and whose third row is B.

Vector Triple Product

Vector product of two vectors can be made to undergo dot or cross product with any third vector.

Scalar Triple Product:-

For three vectors A, B, and C, their scalar triple product is defined as

A.

obtained in cyclic permutation. If , and then is the volume of a parallelepiped having A, B, and C as edges and can easily obtained by finding the determinant of the matrix formed by and .

Vector Triple Product:-

For vectors A, B, and C, we define the vector tipple product as

Note that

not equal to but .

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