Addition Of Vectors Formulas

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1. Vectors:

(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 \(\hat{\mathbf{a}}\). Thus \(\hat{a}=\frac{\vec{a}}{|a|}=\frac{\text { Vector a }}{\text { magnitude of a }}\)

(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.
For example: \(\overrightarrow{\mathrm{AB}}\) , gives position of point B with respect to point
A. Also we can write, \(\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{PB}}-\overrightarrow{\mathrm{PA}} \text { or } \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}\) where O or P are any other points.

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,
If \(\overrightarrow{\mathrm{AB}}=\mathrm{a}, \overrightarrow{\mathrm{BC}}=\mathbf{b}, \text { and } \overrightarrow{\mathrm{AC}}=\mathrm{c}\)

(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.
Thus if \(\overrightarrow{\mathrm{OA}}=a, \overrightarrow{\mathrm{OB}}=\mathbf{b}, \text { and } \overrightarrow{\mathrm{OC}}=\mathrm{c}\)
then \(\overrightarrow{\mathrm{OA}}+\overrightarrow{\mathrm{OB}}=\overrightarrow{\mathrm{OC}}\) i.e. a + b = c

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

If \(\overrightarrow{\mathrm{AB}}\) be any given vector and also suppose that the position vectors of initial point A and terminal point B are a and b respectively, then
\(\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\mathbf{b}-\mathrm{a}\)

5. Distance between two points j

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 \(\overrightarrow{\mathrm{AB}}=\sqrt{\left(\mathrm{x}_{2}-\mathrm{x}_{1}\right)^{2}+\left(\mathrm{y}_{2}-\mathrm{y}_{1}\right)^{2}+\left(\mathrm{z}_{2}-\mathrm{z}_{1}\right)^{2}}\)

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
\(c=\frac{m b+n a}{m+n}\)
Particular cases:

• Any vector along the internal bisector of ∠AOB is given by λ\((\hat{a}+\hat{b})\)
• 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 = \(\mathbf{c}=\frac{m b-n a}{m-n}\)
• If a, b, c are position vectors of vertices of a triangle, then p.v. of its centroid is \(\frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3}\)
• If a, b, c, d are position vectors of vertices of a tetrahedron, then
  … a+b + c + d p.v. of its centroid is \(\frac{a+b+c+d}{4}\)

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

(ii) Three points A, B and C are collinear, if any two vectors \(\overrightarrow{\mathrm{AB}}\), \(\overrightarrow{\mathrm{BC}} \text { and } \overrightarrow{\mathrm{CA}}\) are parallel i.e. one of them is scalar multiple of any one of the remaining vectors.

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
\(\mathbf{a} \| \mathbf{b}\) ⇒ \(\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}\)

(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:
A set of vectors \(\overrightarrow{\mathrm{a}}_{1}, \overrightarrow{\mathrm{a}}_{2}, \ldots . \overrightarrow{\mathrm{a}}_{\mathrm{n}}\) is said to be linearly independent iff
\(x_{1} \vec{a}_{1}+x_{2} \vec{a}_{2}+x_{3} \vec{a}_{3} \ldots+x_{n} \vec{a}_{n}=0\) where x₁ = x₂ = x₃ = …….. = x_n = 0

(v) Linearly dependent vectors:
A set of vector’s \(\vec{a}_{1}, \vec{a}_{2}, \vec{a}_{3} \ldots \vec{a}_{n}\) is said to be linearly dependent iff, there exist scalars x₁, x₂, x₃, …….. x_n not all zero such that \(\mathrm{x}_{1} \overrightarrow{\mathrm{a}}_{1}+\mathrm{x}_{2} \overrightarrow{\mathrm{a}}_{2}+\ldots \ldots+\mathrm{x}_{\mathrm{n}} \overrightarrow{\mathrm{a}}_{\mathrm{n}}=0\)

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