CBSE Class 12 Mathematics Vector Algebra Notes Set 01

BASIC CONCEPTS

1. Vector : Those physical quantities, which are defined by both magnitude and direction are called vector e.g., velocity, acceleration, force, etc.

2. Scalar : Those physical quantities which have only magnitude are called scalar, e.g., area, volume, mass, etc.

3. Position vector : Let \( (x, y, z) \) be a point in space with respect to the origin \( O(0, 0, 0) \). The vector \( \overrightarrow{OP} \) having \( O \) as initial and \( P \) as terminal point is called position vector of \( P \).
Here, position vector of \( P = \overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k} \)

4. Direction cosines : If \( \vec{r} = a\hat{i} + b\hat{j} + c\hat{k} \) makes angle \( \alpha \), \( \beta \), \( \gamma \) with +ve direction of x-axis, y-axis and z-axis respectively, then \( \cos \alpha \), \( \cos \beta \) and \( \cos \gamma \) are the direction cosines of \( \vec{r} \) and are denoted by \( l, m \) and \( n \) where
\( l = \cos \alpha = \frac{a}{\sqrt{a^2 + b^2 + c^2}} \), \( m = \cos \beta = \frac{b}{\sqrt{a^2 + b^2 + c^2}} \)
\( n = \cos \gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \)

5. Direction ratios : If numbers \( a, b, c \) are proportional to direction cosine \( l, m \) and \( n \) respectively of \( \vec{r} \), then \( a, b, c \) are called direction ratios of \( \vec{r} \).

6. Vector joining two points : If \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) are two points, then the vector joining \( A \) and \( B \) is the vector \( \overrightarrow{AB} \) given by
\( \overrightarrow{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} \).
Proof : \( \overrightarrow{AB} = \overrightarrow{AO} + \overrightarrow{OB} \quad \) [By addition of vectors]
\( \overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (x_2\hat{i} + y_2\hat{j} + z_2\hat{k}) - (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) \)
\( = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} \)

7. Components of a vector : If \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) then \( x, y, z \) are called scalar components of \( \vec{r} \) and \( x\hat{i}, y\hat{j}, z\hat{k} \) are called vector components of \( \vec{r} \).

8. If \( \vec{a} \) and \( \vec{b} \) are the position vectors of two points \( A \) and \( B \), then \( \overrightarrow{AB} = \vec{b} - \vec{a} \).

9. If \( \vec{a} = a_1\hat{i} + b_1\hat{j} + c_1\hat{k} \) then \( |\vec{a}| = \sqrt{a_1^2 + b_1^2 + c_1^2} \).

10. If \( \vec{a} \) is any given vector then unit vector in direction \( \vec{a} \), i.e., \( \hat{a} = \frac{\vec{a}}{|\vec{a}|} \).

11. (i) Collinearity of three points : Three points with position vectors \( \vec{a}, \vec{b}, \vec{c} \) are collinear, iff there exist scalars \( x, y, z \) not all zero such that \( x\vec{a} + y\vec{b} + z\vec{c} = \vec{0} \), where \( x + y + z = 0 \).

(ii) Coplanarity of three vectors : Let \( \vec{a} \) and \( \vec{b} \) be two given non-zero non-collinear, vectors. Then any vector \( \vec{r} \), coplanar with \( \vec{a} \) and \( \vec{b} \) can be uniquely expressed as \( \vec{r} = x\vec{a} + y\vec{b} \) for some scalars \( x \) and \( y \).

12. Section formula :
(i) For internal division : The position vector of a point \( C \), which divides internally the line-segment joining two points \( A \) and \( B \) with position vectors \( \vec{a} \) and \( \vec{b} \) in the ratio \( m : n \) (as figure given alongside) is given by
position vector of \( C = \frac{m\vec{b} + n\vec{a}}{m + n} \).
(ii) For external division : The position vector of a point \( C \), which divides externally the line-segment joining two points \( A \) and \( B \) with position vectors \( \vec{a} \) and \( \vec{b} \) in the ratio \( m : n \) (as figure given alongside) is given by
position vector of \( C = \frac{m\vec{b} - n\vec{a}}{m - n} \).

13. Two vectors are said to be orthogonal if they are perpendicular to each other.

14. The dot product (scalar product) of two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \vec{a} . \vec{b} = |\vec{a}||\vec{b}|\cos \theta \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).

15. Properties of dot product of vectors :
(i) \( \vec{a} . \vec{b} = \vec{b} . \vec{a} \) (commutativity)
(ii) \( \vec{a} . (\lambda\vec{b}) = (\lambda\vec{a}) . \vec{b} = \lambda(\vec{a} . \vec{b}) \), \( \lambda \) is a scalar
(iii) \( \vec{a} . (\vec{b} + \vec{c}) = \vec{a} . \vec{b} + \vec{a} . \vec{c} \) (Distributive property)
(iv) \( \vec{a} . \vec{b} = 0 \Leftrightarrow \vec{a} = \vec{0}, \vec{b} = \vec{0} \text{ or } \vec{a} \perp \vec{b} \)
(v) If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then \( \vec{a} . \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \)
(vi) Projection of \( \vec{a} \) on \( \vec{b} = \frac{\vec{a} . \vec{b}}{|\vec{b}|} \) and projection vector of \( \vec{a} \) on \( \vec{b} = \left(\frac{\vec{a} . \vec{b}}{|\vec{b}|^2}\right) . \vec{b} \)
(vii) Projection of \( \vec{b} \) on \( \vec{a} = \frac{\vec{a} . \vec{b}}{|\vec{a}|} \) and projection vector of \( \vec{b} \) on \( \vec{a} = \left(\frac{\vec{a} . \vec{b}}{|\vec{a}|^2}\right) . \vec{a} \)

16. \( \hat{i} . \hat{i} = \hat{j} . \hat{j} = \hat{k} . \hat{k} = 1 \) and \( \hat{i} . \hat{j} = \hat{j} . \hat{k} = \hat{k} . \hat{i} = 0 \), where \( \hat{i}, \hat{j} \) and \( \hat{k} \) are unit vectors along x-axis, y-axis and z-axis respectively.

17. If \( \theta \) is the angle between two vectors \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then
\[ \cos \theta = \frac{\vec{a} . \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{a_1b_1 + a_2b_2 + a_3b_3}{\sqrt{a_1^2 + a_2^2 + a_3^2} \sqrt{b_1^2 + b_2^2 + b_3^2}} \]

18. If \( \vec{a} \perp \vec{b} \), then \( \vec{a} . \vec{b} = 0 \)
\( \implies \) \( a_1a_2 + b_1b_2 + c_1c_2 = 0 \), where \( \vec{a} = a_1\hat{i} + b_1\hat{j} + c_1\hat{k} \) and \( \vec{b} = a_2\hat{i} + b_2\hat{j} + c_2\hat{k} \)

19. The cross product or vector product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \hat{n} \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \) and \( \hat{n} \) is a unit vector perpendicular to the plane of \( \vec{a} \) and \( \vec{b} \) and + ve for a right handed rotation from \( \vec{a} \) to \( \vec{b} \).

20. \( |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \) and \( \sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|} \), where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).

21. Properties of cross product of vectors :
(i) \( \vec{a} \times \vec{b} = - \vec{b} \times \vec{a} \)
(ii) \( \vec{a} \times \vec{a} = \vec{b} \times \vec{b} = \vec{c} \times \vec{c} = \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = \vec{0} \)
(iii) \( \hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i} \) and \( \hat{k} \times \hat{i} = \hat{j} \)
(iv) \( \hat{i} \times \hat{j} = - \hat{j} \times \hat{i}, \hat{j} \times \hat{k} = - \hat{k} \times \hat{j} \) and \( \hat{k} \times \hat{i} = - \hat{i} \times \hat{k} \)
(v) If \( \vec{a} \times \vec{b} = \vec{0} \)
\( \implies \) \( \vec{a} = \vec{0}, \vec{b} = \vec{0} \) or \( \vec{a} \parallel \vec{b} \)
(vi) If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) then \( \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \)

22. Area of a parallelogram, whose sides are represented by \( \vec{a}, \vec{b} \) is \( |\vec{a} \times \vec{b}| \).

23. Unit vector perpendicular to \( \vec{a} \) and \( \vec{b} \) is \( \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} \).

24. Area of parallelogram, whose diagonals are represented by \( \vec{a} \) and \( \vec{b} \) is \( \frac{1}{2} |\vec{a} \times \vec{b}| \).

25. Area of triangle, whose two sides are represented by \( \vec{a} \) and \( \vec{b} \) is given by \( \frac{|\vec{a} \times \vec{b}|}{2} \).

26. If \( \vec{F} \) is a force applied at a point \( A \), then moment of force about the point \( P \) is given by \( |\overrightarrow{AP} \times \vec{F}| \).

27. Cosine formulae : If \( a, b, c \) are lengths of the opposite sides respectively to the angles \( A, B \) and \( C \) of a triangle \( ABC \), then
(i) \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
(ii) \( \cos B = \frac{c^2 + a^2 - b^2}{2ac} \)
(iii) \( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \)

28. Projection formulae : If \( a, b, c \) are lengths of the sides opposite respectively to the angles \( A, B, C \) of a triangle \( ABC \), then
(i) \( a = b \cos C + c \cos B \)
(ii) \( b = c \cos A + a \cos C \)
(iii) \( c = a \cos B + b \cos A \)

29. Lagrange's identity : \( |\vec{a} \times \vec{b}|^2 = \begin{vmatrix} \vec{a} . \vec{a} & \vec{a} . \vec{b} \\ \vec{a} . \vec{b} & \vec{b} . \vec{b} \end{vmatrix} \)
or \( (\vec{a} . \vec{b})^2 + (\vec{a} \times \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2 \)

30. For any two vectors \( \vec{a} \) and \( \vec{b} \), we have
(i) \( |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a} . \vec{b} \)
(ii) \( |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} . \vec{b} \)
(iii) \( |\vec{a} + \vec{b}|^2 + |\vec{a} - \vec{b}|^2 = 2[|\vec{a}|^2 + |\vec{b}|^2] \)
(iv) \( (\vec{a} + \vec{b}) . (\vec{a} - \vec{b}) = |\vec{a}|^2 - |\vec{b}|^2 \)

31. Scalar triple product of vectors :
The scalar triple product of three vectors \( \vec{a}, \vec{b} \) and \( \vec{c} \) denoted by \( [\vec{a}\ \vec{b}\ \vec{c}] \) is equal to the dot product of the first vector by the cross product of remaining two in order.
i.e., \( [\vec{a}\ \vec{b}\ \vec{c}] = \vec{a} . (\vec{b} \times \vec{c}) = \vec{b} . (\vec{c} \times \vec{a}) = \vec{c} . (\vec{a} \times \vec{b}) \)
The scalar triple product is a pseudoscalar (i.e., it reverses sign under inversion).
Since, the cross product of two vectors is calculated by using a determinant as
\[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \]
\[ \vec{a} . (\vec{b} \times \vec{c}) = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) . \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \]
This gives \( [\vec{a}\ \vec{b}\ \vec{c}] = \vec{a} . (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \)
where \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \); \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) and \( \vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k} \)

Properties of scalar triple product :
(i) The scalar triple product of vectors does not change if the order of its factors are circularly rotated, but it changes its sign if they are transposed.
i.e., \( [\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}] \)
(ii) If any vector out of three is equal to any other vector with multiplication of a scalar quantity then the value of scalar triple product is zero.
e.g., let \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \); \( \vec{b} = \lambda a_1\hat{i} + \lambda a_2\hat{j} + \lambda a_3\hat{k} = \lambda \vec{a} \) and \( \vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k} \)
\[ \therefore \quad [\vec{a}\ \vec{b}\ \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ \lambda a_1 & \lambda a_2 & \lambda a_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = \lambda \begin{vmatrix} a_1 & a_2 & a_3 \\ a_1 & a_2 & a_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0 \]
[Note: The value of determinant is zero if any two rows/columns are same.]

Geometrical interpretation of scalar triple product :
Let \( \vec{a}, \vec{b} \) and \( \vec{c} \) be non-zero, non parallel vectors. A parallelepiped is constructed in which three adjacent sides OA, OB and OC are represented by \( \vec{a}, \vec{b} \) and \( \vec{c} \) respectively in magnitude and direction.
Let normal of base OBDC of parallelepiped makes angle \( \theta \) with \( \vec{a} \).
\( \therefore \quad \angle OAM = \theta \), where AM = height of parallelepiped.

Volume of parallelepiped = area of base OBDC × height
\( = |\vec{b} \times \vec{c}| . AM \)
\( = |\vec{b} \times \vec{c}| . OA \cos \theta \quad [\because \text{In } \Delta OAM, \cos \theta = \frac{AM}{OA} \)
\( \implies \) \( AM = OA \cos \theta] \)
\( = |\vec{b} \times \vec{c}| . |\vec{a}| \cos \theta = |\vec{a}| . |\vec{b} \times \vec{c}| \cos \theta \)
\( = \vec{a} . (\vec{b} \times \vec{c}) = [\vec{a}\ \vec{b}\ \vec{c}] = \text{Scalar triple product of } \vec{a}, \vec{b}, \vec{c} \)
[Note: Base OBDC is parallelogram and thus area is \( |\vec{b} \times \vec{c}| \). Also normal vector of base OBDC is \( \vec{b} \times \vec{c} \).]

Coplanarity : Three vectors \( \vec{a}, \vec{b}, \vec{c} \) are coplanar, if the scalar triple product of these three vector is zero, i.e., the volume of parallelepiped so formed is zero and thus it would be flat.
i.e., \( [\vec{a}\ \vec{b}\ \vec{c}] = \vec{a} . (\vec{b} \times \vec{c}) = 0 \Leftrightarrow \vec{a}, \vec{b}, \vec{c} \text{ are coplanar.} \)