The document Section Formula and Scalar Product of Two Vectors JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.

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**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 mid-point 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 PVâ€™s of the vertices A, B, C of a Î”ABC respectively, then the PV of the centroid G of the triangle is a + b + c / 3.

(ii) The PV of incentre of Î”ABC is (BC)a + (CA)b + (AB)c / BC + CA + AB

(iii) The PV of orthocentre of Î”ABC 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 non-zero 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 Î¸, where Î¸ is the angle between the two vectors and 0 < Î¸ < Ï€ .

(i) The angle between two vectors a and b is defined as the smaller angle Î¸ between them, when they are drawn with the same initial point.

Usually, we take 0 < Î¸ < Ï€.Angle between two like vectors is O and angle between two unlike vectors is Ï€ .

(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 Î¸.

(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| cosÎ¸ â‡’ |a|^{2} = a^{2} â‡’ a = |a|

Condition of perpendicularity a * b = 0 <=> a âŠ¥ b, a and b being non-zero vectors.

**Important Points to be Remembered**

(i) (a + b) * (a â€“ b) = |a|^{2}2 â€“ |b|^{2}

(ii) |a + b|^{2} = |a|^{2}2 + |b|^{2} + 2 (a * b)

(iii) |a â€“ b|^{2} = |a|^{2}2 + |b|^{2} â€“ 2 (a * b)

(iv) |a + b|^{2} + |a â€“ b|^{2} = (|a|^{2}2 + |b|^{2}) and |a + b|^{2} â€“ |a â€“ b|^{2} = 4 (a * b)

or a * b = 1 / 4 [ |a + b|^{2} â€“ |a â€“ b|^{2} ]

(v) If |a + b| = |a| + |b|, then a is parallel to b.

(vi) If |a + b| = |a| â€“ |b|, then a is parallel to b.

(vii) (a * b)^{2} â‰¤ |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| = âˆša_{1}^{2} + a_{2}^{2} + a_{3}^{2}

(ix) **Angle between Two Vectors** If Î¸ is angle between two non-zero vectors, a, b, then we have

a * b = |a| |b| cos Î¸

cos Î¸ = 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 Î¸ between a and b is given by

cos Î¸ = a * b / |a| |b| = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} / âˆša_{1}^{2} + a_{2}^{2} + a_{3}^{2} âˆšb_{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.

âˆ´ F * S = dot products of force and displacement.

Suppose F_{1}, F_{1},â€¦, 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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