Plus Two Maths Notes Chapter 10 Vector Algebra is part of Plus Two Maths Notes. Here we have given Plus Two Maths Notes Chapter 10 Vector Algebra.
|Text Book||NCERT Based|
|Chapter Name||Vector Algebra|
|Category||Plus Two Kerala|
Kerala Plus Two Maths Notes Chapter 10 Vector Algebra
- Unit vector: A vector whose magnitude is lo unity.
Unit vector in the direction of
- Vector joining two points.
Let A(x1, y1, z1) and B (x2, y2, z2) be two points, then = Position vector of B Position vector of A
- Direction Cosines.
Let the position vector is of a point
P (x, y, z). make angles α, β, r with the positive direction of x, y & z respectively.
α, β, r are direction angles cosα, cosβ, cosr are direction cosines.
t = cosα = m = cosβ = n = cosr =
cos2α + cos2β + cos2r = 1 i.e., l2 + m2 + n2 = I
the number which are proportional to the direction cosines of a vector is te direction ratios of the vecto . Denoted by a, b, c. a2 + b2 + c2 = l2r2 + m2r2 + n2r2
- Zero vector. Vector whose initial & terminal points coincide.
- Coinitial vector. Two or more vectors having the same initial point.
- Equal vectors. Two vectors and are said to be equal, if they have the same magnitude & direction.
- Collineanty of two vectors.
- Two vectors and are collinear if and only if there exists a non-zero scalar λ such that
- = and
= are collinear if and only if
- Vector addition.
- Triangular law of vector addition
- Parallelogram law of vector addition
- Properties of vector addition:
- (commulative property)
- (Associative property)
- ( is additive identity)
- Two vectors and are parallel, then
= λ, where λ is a scalar.
- Addition and multiplication by a scalar.
Let k and m be any scalars, then
- Collinearity of three points:
Three points A, B, and C are collinear, then
where λ is a scalar.
If and be the position vectors of the points A and B respectively, then the position vector of the point which divides AB in the ratio m: n internally is
The position vector of the mid-point of
Product of Two Vectors
I. Scalar (or dot) product.
where θ is the angle between and , 0 < θ < π
- Geometrically (Projection of on )
- . Projection of b on a =
- Projection of a on b =
- is a real number.
- = 0 ⇔ , where and are two non-zero vectors.
- The angle between two non-zero vectors and is given by cosθ =
- Scalar product in terms of rectangular components:
II Vector (or cross) product.
where θ is the angle between and ,
0 < θ < π, and is the unit vector perpendicular to both and so that
,, form a right handed system.
- Geometrical interpretations:
(1) Area of the parallelogram with adjacent sides and =
(ii) Area of the triangle with adjacent
- is a vector
- = ⇔ , where and are two non-zero vectors.
- The angle between two vectors and is given by, sin θ =
- Vector product in terms of rectangular components
- The vector perpendicular to both and =
- Unit vector perpendicular to the plane of and :
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