The two basic concepts of vector algebra are:

• Position Vector: The vector defining the position of a point A(x, y, z) with respect to the origin O(0, 0, 0) is the position vector of the point A with respect to O. The position vector formula is:
• Direction Cosine of a Vector: In a three-dimensional plane, the direction cosines of a vector are the cosine values of the angles formed by the vector along the three axes. If α, β and ɣ are the angles made by a vector p along the x, y and z axes, then the direction cosines of vector p are (cos α, cos β, cos ɣ), respectively. Direction cosines (or directional cosines) are represented as (l, m, n). Therefore, l = cos α, m = cos β, and n = cos ɣ.

Vectors are of the six following types:

• Zero Vector: A vector with no magnitude or zero distance is called a zero vector. The start and end points of the vector are the same and, therefore, have no direction.
• Unit Vector: A vector whose length or magnitude is equal to 1 is a unit vector. If we divide a vector by its magnitude, it will give us a unit vector.
• Coinitial Vectors: Coinitial vectors are vectors with the same starting point. For example, vectors PQ, PR, and PS are coinitial vectors since they have the same starting point P.
• Collinear Vectors: Vectors that are parallel to the same line, irrespective of their magnitude and direction, are collinear vectors. Vectors A(x1, y1, z1) and B(x2, y2, z2) are collinear if .
• Equal Vectors: Vectors that have the same magnitude and direction as each other but can have different start and end points are known as equal vectors.
• Negative of a Vector: If two vectors have the same magnitude, but different directions from each other, then one is called the negative of the other vector.

There are two laws of addition of vectors:

• Triangle Law of Vector Addition: According to this law, the initial point of one vector and the terminating point of the other one should be the same.

PR = PQ + QR

• Parallelogram Law of Vector Addition: According to this law, the addition of two vectors p and q, representing adjacent sides of a parallelogram, results in the diagonal of the parallelogram.

OP + OQ = OR

There are three properties of vector addition:

• Commutative Property: This property says that the addition of two or more vectors provides the same result, regardless of the addition order of the vectors.

p + q = q + p

• Associative Property: The associative property states that the addition of three or more vectors gives the same result, irrespective of how the vectors are grouped, i.e. the sum of p and q when added with r gives the same result as when p is added to the sum of q and r.

(p + q) + r = p + (q + r)

• Additive Identity: The additive identity property states that if a given vector is added with the zero vector, then the result is the same as the given vector. Therefore, zero vector is the additive identity for the addition of vectors.

p + 0 = 0 + p = p

In multiplication of a vector by a scalar quantity, the magnitude of the vector changes depending on the magnitude of the scalar, say ɑ, and the direction of the vector remains the same if ɑ is positive and changes in the opposite direction if ɑ is negative.

ɑ|p| = |ɑp|

If ɑ = -1, then ɑ|p| = -p

Then p + (-p) = 0. Therefore, -p is the negative or additive inverse of p.

When a vector, let’s say r in a three-dimensional plane, is resolved along each of the three axes, the result will be the component of the vector along each axis.

Therefore, , where x, y and z are the scalar components of  and  and  are the vector components of along the respective axes.

If and are any two points in a three-dimensional plane, then the vector is the vector joining the two points and .

To get the magnitude of the vector , the triangle law of vector addition can be applied:

Therefore, the magnitude of

Section formula is used to find the position vector OP of a point P that divides a line segment joining the points M and N, with position vectors m and n respectively, internally or externally, in a particular ratio a:b.

For internal division, the section formula is: OP = (bm + an)/(a + b).

For external division, the section formula is: OP = (bm - an)/(m - n).

Note: Position vector of the midpoint of the line segment joining points M(m) and N(n) is given by OP = (m + n)/2.

There are two methods to find the product of two vectors. First method is the dot product or scalar multiplication of two vectors; this type of multiplication results in a scalar quantity. The second type of multiplication is the vector product of two vectors; this type of multiplication results in a vector quantity.

The scalar or dot product of two vectors is the product of their magnitudes and the cosine of the angle formed by the vectors. If there are two vectors p and q, with respective magnitudes |p| and |q|, then the dot product of these vectors is: p • q = |p||q| cos θ , where θ is the angle formed between the two vectors. If either of the vectors is zero, then the resultant product is also zero.

The algebraic formula for the dot product of two vectors and .

The following are some properties of scalar product of two vectors:

• Commutative property: p • q = q • p = |p||q| cos θ
• Orthogonal Property: If p • q = 0, then either one of the vectors is zero, or cos θ = 0 ⇒ θ = π/2, which means that they are perpendicular to each other.
• Scalar Multiplication Property: (ap) • (bq) = (aq) • (bp) = ab p • q
• Distributive Property: p • (q + r) = p • q + p • r

If a vector PQ forms an angle θ with a given directed line l, in the anticlockwise direction, then the projection of vector PQ on line l is a vector p with magnitude |PQ| cos θ.

The direction of  and that of the line l is the same if cos θ is positive. If cos θ is negative, the direction is opposite to that of the line.

If there are two vectors p and q then the vector or cross product of p and q is r.

r = p × q.

Therefore, the magnitude of r = pq cos θ, where θ is the angle between p and q and the direction of r is perpendicular to both p and q. To find the direction of this vector, we use the ‘right-hand thumb rule’, in which, when you curl your fingers in the direction from p to q, the direction that your thumb points in is the direction of r, the cross product.

Let’s look at some of the properties of the cross product of two vectors:

• Vector product is not commutative: p × q ≠ q × p
• Distributive Property: p × (q + r) = p × q + p × r
• kp × q = k(p × q) = p × kq
• If the vectors are collinear, then their cross product is zero

1 . What are the important lessons of Class 12 Maths Chapter 10?

Some of the important topics are:

• Basic concepts
• The types of vectors: zero, unit, collinear and coinitial vectors the negative of a vector
• Position vector and direction cosines
• Addition and properties of addition of vectors
• Components of a vector
• Scalar and vector product of vectors and their respective properties

2 . What is the difference between scalar and vector?

A scalar quantity has only magnitude, while a vector quantity has magnitude and direction both.

3 . How can the NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra be used for exams?

Here are some of the ways in which the NCERT Solutions can be used as a tool to prepare for Class 12 Maths exams:

• The answers to the problems are detailed and provided by subject matter experts in an easy-to-follow manner.
• The solutions guide has a list of all the important formulas required in the chapter.
• The solutions help students understand the Class 12 CBSE exam question patterns, marks distribution and more.
• All the important topics under the chapter ‘Vector Algebra', such as types of vectors, properties of vector addition, dot product and cross product of two vectors etc. are thoroughly covered.