Vector Algebra Class 12 Notes - Chapter 10

Position of a Vector

If we are provided with a point Q(x,y,z) and

\(\begin{array}{l}\overline{OQ}(=\underset{r}{\rightarrow})=x\widehat{i}+y\widehat{j}+z\widehat{k}\end{array} \)
and the magnitude is given by  
\(\begin{array}{l}\sqrt{x^{2}+y^{2}+z^{2}}\end{array} \)
. The direction ratios for a vector are its scalar components and are responsible for its projections along the respective axes.

The relation between magnitude, direction ratios, and direction cosines of a vector

If a vector has been given with dimensions such as magnitude(p), direction ratios (x,y,z) and direction cosines (l,m,n) then the relation between them is:

l=x/p, m=y/p,n=z/p

The order taken for the vector sum of the three sides of the triangle is

\(\begin{array}{l}\underset{0}{\rightarrow}\end{array} \)

And the vector sum of coinitial vectors is the diagonal of the parallelogram, which has the vectors as its adjacent sides.

When multiplying a vector by a scalar λ, the magnitude of the vector changes by the multiple |λ|, and the direction remains the same (or makes it opposite) according to the value of λ being positive (or negative).

Position Vector of a Point R on a Line Segment

A line segment joining two points, such as P having a position vector of

\(\begin{array}{l}\underset{a}{\rightarrow}\end{array} \)
and Q having a position vector of
\(\begin{array}{l}\underset{b}{\rightarrow}\end{array} \)
is divided by a point R by the ratio m;n then,
  • The internal ratio is given by
    \(\begin{array}{l}n\underset{a}{\rightarrow}+m\underset{b}{\rightarrow}/m+n\end{array} \)
  • The external ratio is given by
\(\begin{array}{l}m\underset{b}{\rightarrow}-n\underset{a}{\rightarrow}/m-n\end{array} \)

The scalar product of two position vectors

\(\begin{array}{l}\underset{a}{\rightarrow}\end{array} \)
and
\(\begin{array}{l}\underset{b}{\rightarrow}\end{array} \)
and subtending an angle
\(\begin{array}{l}\theta\end{array} \)
is represented by
\(\begin{array}{l}\underset{a}{\rightarrow}\end{array} \)
.
\(\begin{array}{l}\underset{b}{\rightarrow}\end{array} \)
=
\(\begin{array}{l}\left | \underset{a}{\rightarrow} \right |\left | \underset{b}{\rightarrow} \right |cos\theta\end{array} \)

We can find the value of

\(\begin{array}{l}\theta\end{array} \)
from the above equation.
Also Access 
NCERT Solutions for Class 12 Maths Chapter 10
NCERT Exemplar for Class 12 Maths Chapter 10

Important Questions

  1. Find the ratio in which B divides AC and also show that points A(2, – 3, – 9), B(6, 0, –1), and C(10, 4, 6) are not collinear.
  2. Determine the girl’s displacement from her initial point of departure if she walks 5 km towards the West, then walks 4 km in a direction 40° East of the North and stops.
  3. Draw a unit vector in the XY-plane, making an angle of 30° with the positive direction of the x-axis.
  4. Find the area of the triangle with vertices P(2, 3, 4), Q(3, 4, 6) and R(2, 6, 7).
  5. Show that the points P(1, 2, 7), Q(2, 6, 3) and R(3, 10, –1) are collinear

Also, Read

Algebra Algebra Formulas

Frequently Asked Questions on CBSE Class12 Maths Notes Chapter 10 Vector Algebra

Q1

What is a constant?

A constant is a quantity which has a fixed value.

Q2

What are like and unlike terms?

In algebra, terms are terms that have the same variables and powers. Unlike terms are two or more terms that are not like terms, i.e. they do not have the same variables or powers.

Q3

What is a vector quantity?

Any quantity that has both magnitude and direction but not position is known as a vector quantity.

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