(a) Definition: A directed line segment is called vector, picture shows a vector.
“AB” is magnitude of vector and its direction is from A to B. Imp.: Magnitude of vector is a scalar quantity.
(b) Types of Vectors
(i) Zero or null vector:
A vector whose magnitude is zero is called zero or null vector.
(ii) Unit vector:
A vector of unit magnitude is called a unit vector. A unit vector in the direction of a is denoted by a^. Thus
(iii) Equal vector:
Two vectors a and b are said to be equal, if |a| = |b| & they have the same direction.
(iv) Co-initial vector’s:
Vector having same initial point.
(v) Free vector’s:
The vector whose location is not fixed.
Imp.: All vector’s we consider in this topic are free vector’s.
(vi) Position vector:
A vector which give position of one point with respect to another is called position vector.
2. Addition of Vectors
(i) Triangle law of addition:
If two vectors are represented by two consecutive sides of a triangle then their sum is represented by the third side of the triangle but in opposite direction. This is known as the triangle law of addition of vectors. Thus,
(ii) Parallelogram Law of Addition:
If two vectors are represented by two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram.
Where OC is a diagonal of the parallelogram OABC.
(iii) Addition in component form :
If a = a1i + a2 j + a3k and b = b1i + b2j + b3k
then their sum is defined as a + b = (a1 + b1)i + (a2 + b2j + (a3 + b3)k.
3. Subtraction of vectors
If a and b are two vectors, then their subtraction a – b is defined as a – b = a + (- b)
where – b is the negative of b having magnitude equal to that of b and direction opposite to b.
4. Vectors in terms of position vectors of end points
Let A and B be two given points whose coordinates are respectively (x1, y1, z1) and (x2, y2, z2). Distance between the points A and |
B = magnitude of
6. Multiplication of a vector by a scalar
If a is a vector and m is a scalar (i.e. a real number) then ma is a vector NOTES whose magnitude is m times that of a and whose direction is the same as that of a, if m is positive and opposite to that of a, if m is negative,
∴ magnitude of ma = m |a|
Again if a = a1 + a2 j + a3k then ma = (ma1)i + (ma2)j + (ma3)k
7. Position Vector of a dividing point
If a and b are the position vectors of two points A and B, then the position vector c of a point P dividing AB in the ratio m : n is given by
If the point P divides AB in the ratio m : n externally, then m/n will be negative. If m is positive and n is negative, then p.v. c of P is given by c =
8. Collinearity of three points
(i) If a, b, c be position vectors of three points A,B and C respectively and x, y, z be three scalars so that all are not zero, then the necessary and sufficient conditions for three points to be collinear is that xa + yb + zc = 0 and x + y + z = 0
9. (a) Relation between two parallel vectors
If a = a1i + a2j + a3k and b = b1i + b2j + b3k then from the property of parallel vector, we have
(b) Relation between perpendicular vector’s a1b1 + a2b2 + a3b3 = 0
10. Coplanar & non-coplanar vector
(i) If a, b, c be three coplanar vectors, then a vector c can be expressed uniquely as linear combination of remaining two vectors i.e. c = λa + μb, where λ and μ are suitable scalars. Again c = λa + μb ⇒ vectors a, b and c are coplanar.
If a, b, c be three coplanar vectors, then there exist three non zero scalars x, y, z so that xa + yb + zc = 0
(ii) If a, b, c be three non coplanar non zero vector then xa + yb + zc = 0 ⇒ x = 0, y = 0, z = 0
(iii) Any vector r can be expressed uniquely as the linear combination of three non coplanar and non-zero vectors a, b and c i.e. r = xa + yb + zc, where x, y and z are scalars.
(iv) Linearly independent vectors:
✅ 12th grade math formulas ⭐️⭐️⭐️⭐️⭐