Section Formula
Let A and B be two points with position vectors a and b, respectively and OP= r.
(i) Let P be a point dividing AB internally in the ratio m : n. Then,
r = m b + n a / m + n
Also, (m + n) OP = m OB + n OA
(ii) The position vector of the midpoint of a and b is a + b / 2.
(iii) Let P be a point dividing AB externally in the ratio m : n. Then,
r = m b + n a / m + n
Position Vector of Different Centre of a Triangle
(i) If a, b, c be PV’s of the vertices A, B, C of a ΔABC respectively, then the PV of the centroid G of the triangle is a + b + c / 3.
(ii) The PV of incentre of ΔABC is (BC)a + (CA)b + (AB)c / BC + CA + AB
(iii) The PV of orthocentre of ΔABC is
a(tan A) + b(tan B) + c(tan C) / tan A + tan B + tan C
Scalar Product of Two Vectors
If a and b are two nonzero vectors, then the scalar or dot product of a and b is denoted by a * b and is defined as a * b = a b cos θ, where θ is the angle between the two vectors and 0 < θ < π .
(i) The angle between two vectors a and b is defined as the smaller angle θ between them, when they are drawn with the same initial point.
Usually, we take 0 < θ < π.Angle between two like vectors is O and angle between two unlike vectors is π .
(ii) If either a or b is the null vector, then scalar product of the vector is zero.
(iii) If a and b are two unit vectors, then a * b = cos θ.
(iv) The scalar product is commutative
i.e., a * b= b * a
(v) If i , j and k are mutually perpendicular unit vectors i , j and k, then
i * i = j * j = k * k =1
and i * j = j * k = k * i = 0
(vi) The scalar product of vectors is distributive over vector addition.
(a) a * (b + c) = a * b + a * c (left distributive)
(b) (b + c) * a = b * a + c * a (right distributive)
Note Length of a vector as a scalar product
If a be any vector, then the scalar product
a * a = a a cosθ ⇒ a^{2} = a^{2} ⇒ a = a
Condition of perpendicularity a * b = 0 <=> a ⊥ b, a and b being nonzero vectors.
Important Points to be Remembered
(i) (a + b) * (a – b) = a^{2}2 – b^{2}
(ii) a + b^{2} = a^{2}2 + b^{2} + 2 (a * b)
(iii) a – b^{2} = a^{2}2 + b^{2} – 2 (a * b)
(iv) a + b^{2} + a – b^{2} = (a^{2}2 + b^{2}) and a + b^{2} – a – b^{2} = 4 (a * b)
or a * b = 1 / 4 [ a + b^{2} – a – b^{2} ]
(v) If a + b = a + b, then a is parallel to b.
(vi) If a + b = a – b, then a is parallel to b.
(vii) (a * b)^{2} ≤ a^{2}2 b^{2}
(viii) If a = a_{1}i + a_{2}j + a_{3}k, then a^{2} = a * a = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}
Or
a = √a_{1}^{2} + a_{2}^{2} + a_{3}^{2}
(ix) Angle between Two Vectors If θ is angle between two nonzero vectors, a, b, then we have
a * b = a b cos θ
cos θ = a * b / a b
If a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k
Then, the angle θ between a and b is given by
cos θ = a * b / a b = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} / √a_{1}^{2} + a_{2}^{2} + a_{3}^{2} √b_{1}^{2} + b_{2}^{2} + b_{3}^{2}
(x) Projection and Component of a Vector
Projection of a on b = a * b / a
Projection of b on a = a * b / a
Vector component of a vector a on b
Similarly, the vector component of b on a = ((a * b) / a^{2}) * a
(xi) Work done by a Force
The work done by a force is a scalar quantity equal to the product of the magnitude of the force and the resolved part of the displacement.
∴ F * S = dot products of force and displacement.
Suppose F_{1}, F_{1},…, F_{n} are n forces acted on a particle, then during the displacement S of the particle, the separate forces do quantities of work F_{1} * S, F_{2} * S, F_{n} * S.
Total workdone is
Here, system of forces were replaced by its resultant R.
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