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# Vector Algebra class 12 Notes Mathematics

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## Vector Algebra Class 12 Notes Mathematics

Download CBSE class 12th revision notes for chapter 10 Vector Algebra in PDF format for free. Download revision notes for Vector Algebra class 12 Notes and score high in exams. These are the Vector Algebra class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 10 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.

## CBSE Class 12 Mathematics Revision Notes Chapter 10 Vector Algebra

Vector: A quantity that has magnitude as well as direction is called vector.

• Zero Vector: A vector whose intial and terminal point coincide is called a zero vector or a null vector. It is denoted as $\overrightarrow {\rm{O}}$.
• Co-initial vectors: Two or more vectors having the same initial points are called co-initial vectors.
• Collinear vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.
• Equal vectors: Two vectors are said to be equal, if they have the same magnitude and direction regardless of the position of their initial points.
• Negative of a vector: A vector whose magnitude is the same as that of a given vector, but direction is opposite to that of it, is called negative of the given vector.
• Position vector of a point P (x, y) is given as $\overrightarrow {{\rm{OP}}} \left( { = \overrightarrow r } \right) = x\widehat i + y\widehat j + z\widehat k$  and its magnitude by
• The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
• The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as:
• The vector sum of the three sides of a triangle taken in order is  $\overrightarrow {\rm{O}}$
• The vector sum of two conidial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
• The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction same (or makes it opposite) according as the value of λ is positive (or negative).
• For a given vector $\overrightarrow a$, the vector $\widehat a = {{\overrightarrow a } \over {\left| {\overrightarrow a } \right|}}$gives the unit vector in the direction of ${\overrightarrow a }$
• The position vector of a point R dividing a line segment joining the points P and Q whose position vectors are  ${\overrightarrow a }$ and $\overrightarrow b$respectively, in the ratio
(i)  internally, is given by ${{n\overrightarrow a + m\overrightarrow b } \over {m + n}}$
(ii)  externally, is given by ${{m\overrightarrow b - n\overrightarrow a } \over {m - n}}$
• The scalar product of two given vectors ${\overrightarrow a }$ and $\overrightarrow b$ having angle θ between them is defined as $\overrightarrow a .\overrightarrow b = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\cos \theta$
Also, when $\overrightarrow a .\overrightarrow b$is given, the angle between the vectors ${\overrightarrow a }$ and $\overrightarrow b$ may be determined by $\cos \theta = {{\overrightarrow a .\overrightarrow b } \over {\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}}$
• If  is the angle between two vector  ${\overrightarrow a }$ and $\overrightarrow b$,  then their cross product is given as
$\overrightarrow a \times \overrightarrow b = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\sin \theta .\widehat n$whereis a unit vector perpendicular to the plane containing ${\overrightarrow a }$ and $\overrightarrow b$. Such that $\overrightarrow a ,\overrightarrow b ,\widehat n$ form right handed system of coordinate axes.
• If we have two vectors  ${\overrightarrow a }$ and $\overrightarrow b$ given in component form as $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$  and  $\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k$  and $\lambda$ be  any scalar, then,
$\overrightarrow a + \overrightarrow b = \left( {{a_1} + {b_1}} \right)\widehat i + \left( {{a_2} + {b_2}} \right)\widehat j + \left( {{a_3} + {b_3}} \right)\widehat k$
$\lambda \overrightarrow a = \left( {\lambda {a_1}} \right)\widehat i + \left( {\lambda {a_2}} \right)\widehat j + \left( {\lambda {a_3}} \right)\widehat k$
$\overrightarrow a .\overrightarrow b = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}$

Parallelogram Law of vector addition: If two vectors ${\overrightarrow a }$ and $\overrightarrow b$ are represented by adjacent sides of a parallelogram in magnitude and direction, then their sum $\overrightarrow a + \overrightarrow b$ is represented in magnitude and direction by the diagonal of the parallelogram through their common initial point. This is known as Parallelogram Law of vector addition.

## CBSE Class 12 Revision Notes and Key Points

Vector Algebra class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math’s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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