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