# Ch10. Vector Algebra

## Achievements

The following badges can be earned while learning.

## Certificate

Certificate on successful completion of this course.

THIS IS A COMBO OF TWO COURSES

COURSE – 1

In this online course, you will learn vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors. This course also has solutions of NCERT Exercise and a few important examples. Please note, some of the part of this course comprises of live session recordings of Ashish Kumar (Agam Sir). Please ignore, if you have heard Agam Sir discussing about live session or doubts session in the course. Live sessions or doubts sessions are no longer offered.

## Course Content

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COURSE – 2

This online course is an extended part of Ch10. Vector Algebra Class 12 Maths. The course is based on the assignments by Ashish Kumar (Agam Sir), which have questions from NCERT Exemplar, Board’s Question Bank, R. D. Sharma etc. The PDF of assignments can be downloaded within the course.

## Course Content

The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

Question 1. A vector {\displaystyle \overrightarrow r } is inclined to x – axis at {\displaystyle 45^\circ } and y – axis at {\displaystyle 60^\circ } , if {\displaystyle |\vec r|\,\, = 8\,{\rm{units}}} , then find {\displaystyle \vec r} .
Question 2. If {\displaystyle |\vec a + \vec b|\, = 60,\,\,|\vec a - \vec b|\, = \,40} and {\displaystyle \left| {\vec b} \right| = 46} find {\displaystyle |\vec a|} .
Question 3. If {\displaystyle {(\vec a \times \vec b)^2} + {(\vec a.\,\vec b)^2} = 144} and {\displaystyle |\vec a|\, = \,4} . Find the value of {\displaystyle |\vec b|} .
Question 4. If {\displaystyle \vec a,\,\,\vec b} are two vectors such that {\displaystyle |\vec a + \vec b|\, = \,|\vec a|} then prove that {\displaystyle 2\vec a + \vec b} is perpendicular to {\displaystyle \vec b} .
Question 5. The points A, B and C with position vectors {\displaystyle 3\hat i - y\hat j + 2\hat k,\,\,5\hat i - \hat j + \hat k} and {\displaystyle 3x\hat i + 3\hat j - \hat k} are collinear. Find the values of x and y. Further, find the ratio in which the point B divides AC.
Question 6. If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is {\displaystyle \sqrt 3 } .
Question 7. If {\displaystyle \vec a\,\,{\rm{and}}\,\,\vec b} are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.
Question 8. Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
Question 9. A vector {\displaystyle \vec r} is inclined at equal angles to the three axes. If the magnitude of {\displaystyle \vec r} is {\displaystyle 2\sqrt 3 } units, find {\displaystyle \vec r} .
Question 10. A vector {\displaystyle \vec r} has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of {\displaystyle \vec r} , given that {\displaystyle \vec r} makes an acute angle with x – axis.
Question 11. Find a vector of magnitude 6, which is perpendicular to both the vectors {\displaystyle 2\hat i - \hat j + 2\hat k\,\,{\rm{and}}\,\,4\hat i - \hat j + 3\hat k} .

### Assignment – 2

Question 1. If {\displaystyle \hat a } and {\displaystyle \hat b} are unit vectors inclined at an angle {\displaystyle \theta } then prove that
a. {\displaystyle \cos \frac{\theta }{2} = \frac{1}{2}|\hat a + \hat b|}
b. {\displaystyle \tan \frac{\theta }{2} = \left| {\frac{{\hat a - \hat b}}{{\hat a + \hat b}}} \right|}
Question 2. For any vector {\displaystyle \vec a} prove that {\displaystyle |\vec a \times \hat i{|^2} + |\vec a \times \hat j{|^2} + |\vec a \times \hat k{|^2} = 2|\vec a{|^2}} .
Question 3. Show that {\displaystyle {(\vec a \times \vec b)^2} = \,|\vec a{|^2}\,|\vec b{|^2} - {(\vec a.\,\vec b)^2} = {\begin{vmatrix}{\vec a.\,\vec a}&{\vec a.\,\vec b}\\ {\vec a.\,\vec b}&{\vec b.\,\vec b} \end{vmatrix}}}
Question 4. Let {\displaystyle \vec a,\,\,\vec b\,\,{\rm{and}}\,\,\vec c} be unit vectors such that {\displaystyle \vec a.\,\vec b = \vec a.\,\vec c = 0} and the angle between {\displaystyle \vec b\,\,{\rm{and}}\,\,\vec c} is {\displaystyle \frac{\pi }{6}} , prove that {\displaystyle \vec a = \pm \,2\,(\vec b \times \vec c)} .
Question 5. If {\displaystyle \vec a,\,\,\vec b\,\,{\rm{and}}\,\,\vec c} are three vectors such that {\displaystyle \vec a + \vec b + \vec c = \vec 0} , then prove that {\displaystyle \vec a \times \vec b = \vec b \times \vec c = \vec c \times \vec a} .
Question 6. Let {\displaystyle \vec a,\,\,\vec b\,\,{\rm{and}}\,\,\vec c} be three non-zero vectors such that {\displaystyle \vec c} is a unit vector perpendicular to both {\displaystyle \vec a\,\,{\rm{and}}\,\,\vec b} . If the angle between {\displaystyle \vec b\,\,{\rm{and}}\,\,\vec a} is {\displaystyle \frac{\pi }{6}} , then prove that {\displaystyle {[\vec a\,\,\,\vec b\,\,\,\vec c]^{\,2}} = \frac{1}{4}|\vec a{|^{\,2}}\,|\vec b{|^{\,2}}} .
Question 7. If {\displaystyle \vec a \times \vec b = \vec c \times \vec d\,\,{\rm{and}}\,\,\vec a \times \vec c = \vec b \times \vec d} , prove that {\displaystyle (\vec a - \vec d)} is parallel to {\displaystyle (\vec b - \vec c),\,\,{\rm{where}}\,\,\vec a \ne \vec d\,\,{\rm{and}}\,\vec b \ne \vec c} .
Question 8. Find a vector of magnitude {\displaystyle \sqrt {171} } which is perpendicular to both of the vectors {\displaystyle \vec a = \hat i + 2\hat j - 3\hat k\,\,{\rm{and}}\,\,\vec b = 3\hat i - \hat j + 2\hat k} .
Question 9. If {\displaystyle \vec a = \hat i + \hat j + \hat k\,\,{\rm{and}}\,\,\vec b = \hat j - \hat k} , find a vector {\displaystyle \vec c} such that {\displaystyle \vec a \times \vec c = \vec b\,\,{\rm{and}}\,\,\vec a.\,\vec c = 3} .
Question 10. If {\displaystyle \vec a = 2\hat i - \hat j + \hat k,\,\,\vec b = \hat i + \hat j - 2\hat k\,\,{\rm{and}}\,\,\vec c = \hat i + 3\hat j - \hat k} , find {\displaystyle \lambda } such that {\displaystyle \vec a} is perpendicular to {\displaystyle \lambda \vec b + \vec c} .

Syllabus medium English Hinglish (Hindi + English) 12 Online learning Self-learning from videos Mathematics