## Unit 03: Vectors (Solutions)

This is a third unit of the book Mathematics 11 published by Khyber Pakhtunkhwa Textbook Board, Peshawar, Pakistan. On this page we have provided the solutions of the questions.

After reading this unit the students will be able to

- Define a scalar and a vector.
- Give geometrical representation of a vector.
- Give the following fundamental definitions using geometrical representation.
- magnitude of a vector,
- equal vectors,
- negative of a vector,
- unit vector,
- zero/null vector,
- position vector,
- parallel vectors,
- addition and subtraction of vectors,
- triangle, parallelogram and polygon laws of addition,
- scalar multiplication.

- Represent a vector in a Cartesian plane by defining fundamental unit vectors $i$ and $j$.
- Recognize all above definitions using analytical representation.
- Find a unit vector in the direction of another given vector.
- Find the position vector of a point which divides the line segment joining two points in a given ratio.
- Use vectors to prove simple theorems of descriptive geometry.
- Recognize rectangular coordinate system in a space.
- Define unit vectors $i$, $j$ and $k$.
- Recognize components of a vector.
- Give analytic representation of a vector.
- Find magnitude of a vector.
- Repeat all fundamental definitions for vectors in space which, in the plane, have already been discussed.
- State and prove
- commutative law for vector addition.
- associative law for vector addition.

- Prove that:
- $O$ as the identity for vector addition.
- $-A$ as the inverse for $A$ .

- State and prove:
- commutative law for scalar multiplication,
- associative law for scalar multiplication,
- distributive laws for scalar multiplication.

- Define dot or scalar product of two vectors and give its geometrical interpretation.
- Prove that.
- $i.i=j.j=k.k=1$

- $i.j=j.k=k.i=0$
- Express dot product in terms of components.
- Find the condition for orthogonality of two vectors.
- Prove the commutative and distributive laws for dot product.
- Explain direction cosines and direction ratios of a vector.
- Prove that the sum of the squares of direction cosines is unity.
- Use dot product to find the angle between two vectors.
- Find the projection of a vector along another vector.
- Find the work done by a constant force in moving an object along a given vector.
- Define cross or vector product of two vectors and give its geometrical interpretation.
- Prove that:
- $i\times i =j\times j =k\times k=0$,
- $i\times j = k$,
- $j\times k =k\times j = i$,

- Express cross product in terns of components.
- Prove that the magnitude of $A \times B$ represents the area of a parallelogram with adjacent sides $A$ and $B$ .
- Find the condition for parallelism of two non-zero vectors.
- Prove the distributive laws for cross product.
- Use cross product to find the angle between two vectors.
- Find the vector moment of a given force about a given point.
- Define scalar triple product of vectors.
- Express scalar triple product of vectors in terms of components (determinantal form).
- Prove that:
- $i.j\times k =j.k\times i=k.i\times j=1$,
- $i.k\times j = J.i\times k=k.j\times i=-1$

- Prove that dot and cross are inter-changeable in scalar triple product.
- Find the volume of
- a parallelepiped,
- a tetrahedron, determined by three given vectors.

- Define coplanar vectors and find the condition for coplanarity of three vectors.