all (A to Z) formula of vector for class 11

Question

Aman Bansal · Answer

Dear Shubham,

The formula for the length of a 2D vector is the Pythagorean Formula. Say that the vector is represented by (x, y)^T. Put the vector with its tail at the origin. Now make a triangle by drawing the two sides:

side_1   =  (x, 0)^T

side_2   = (0, y)^T.

The length of side_1 is x, and the length of side_2 is y, so:

length (x, y)^T = ( x² + y² )

In this formula, means the positive square root. We don''t (of course) want the length to be negative.

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Aman Bansal · Answer

Dear Shubham,

vector quantity has both magnitude and direction. Acceleration, velocity, force and displacement are all examples of vector quantities. A scalar quantity is has only magnitude (so the direction is not important). Examples include speed, time and distance.

Unit Vectors

A unit vector is a vector which has a magnitude of 1. There are three important unit vectors which are commonly used and these are the vectors in the direction of the x, y and z-axes. The unit vector in the direction of the x-axis is i, the unit vector in the direction of the y-axis is j and the unit vector in the direction of the z-axis is k.

Writing vectors in this form can make working with vectors easier.

The Magnitude of a Vector

The magnitude of a vector can be found using Pythagoras''s theorem.

The magnitude of ai + bj = √(a² + b²)

We denote the magnitude of the vector a by | a |

Position Vectors

Position vectors are vectors giving the position of a point, relative to a fixed point (the origin).

For example, the points A, B and C are the vertices of a triangle, with position vectors a, b and c respectively:

You can draw in the origin wherever you want.

Notice that = - a + b = b - a because you can get from A to B by going from A to O and then going from O to B.

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Sorabh Gulia · Answer

Vector Formulas A vector can also be defined as an element of a vector space. Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector, an n-dimensional vector is often called an n-vector, and so on. The important formulas of vectors are given below: 1. The position vector of any point p(x,y) is or OP = ( x,y ). 2.The magnitude of position vector and direction 3. The unit vector = where the magnitude of unit vector is 1 Or,the unit vector = 4.The two vectors and are parallel if and where k and m are the scalars. 5.If then is the result vector which is the triangle law of vector addition. 6. The scalar or dot product of any two vectors . 7. The angle between two vectors is 8. and , then : where 9. If the position vector of A is , position vector of point B is and position vector of mid-point M is m then 10. If the point P divides Ab internally in the ratio m:n then position vector of P is given by which is a section formula. 11.If P divides AB externally in the ratio m:n then PRODUCT OF TWO VECTORS 1 .Scalar Product ( dot product ) Let then dot product of & is devoted by read as dot and defined by Note: if OR The scalar product of & is devoted by , where being angle between & Note:1 Note:2 & are perpendicular if = i.e or 2 .Properties of Scalar Product i. . ii. . iii. iv. v. If then 3. Vector (cross) Product of two vectors. Let be two vectors then the cross product of is devoted by and defined by = We can define in terms of determinants as follows = Note:1 being angle between & Note:2 Note:3 If , the and & are parallel if . 4. Properties of cross product i. ii. iii. iv. v. is perpendicular to both and vi. is a Area of paralelogram with sides and vii. = area of triangle having , , as position vectors of vertices of a triangle.