01_Ex 7.1

Exercise Questions

Questions	Links
Q1. Write the vectors $\mathbf{P Q}$ in the form $\mathbf{x i}+\mathbf{y j}$ …	7.1 Q-1
Q2. Find the magnitude of vector of $U$ …	7.1 Q-2
Q3. If $\underline{U}=2 \underline{\mathbf{i}}-7 \underline{j}, \underline{V}=\underline{\mathbf{i}}-6 \underline{j}$ and $\underline{\mathrm{W}}=-\underline{\mathbf{i}}+\underline{\mathbf{j}}$ …	7.1 Q-3
Q4. Find the sum of vectors $A \vec{B}$ and $\vec{C D}$ …	7.1 Q-4
Q5. Find the vector from the point $A$ to…	7.1 Q-5
Q6. Find a unit vector in the direction of…	7.1 Q-6
Q7. If $A, B, C$ are respectively the points…	7.1 Q-7
Q8. If $B, C$ and $D$ are respectively…	7.1 Q-8
Q9. If $O$ is the origin and $O \vec{P}=A \vec{B}$ …	7.1 Q-9
Q10. Use vectors to show that $A B C D$ is a parallelogram…	7.1 Q-10
Q11. If $A \vec{B}=C\vec{D}$ . Find the…	7.1 Q-11
Q12. Find the position vector of the point of…	7.1 Q-12
Q13. Prove that line segment joining the midpoint…	7.1 Q-13
Q14. Prove that line segments joining the mid-point..	7.1 Q-14

Overview

This exercise focuses on the foundational concept of vectors, their operations, and applications in geometry. Vectors play a crucial role in understanding the direction and magnitude of quantities in various fields such as physics, engineering, and computer science. The exercise will explore vector addition, multiplication, unit vectors, and applications in geometric problems like parallelograms and triangle properties.

Key Concepts

Vectors: Quantities with both magnitude and direction.
Unit Vector: A vector with magnitude equal to 1, used to indicate direction.
Position Vectors: Vectors that represent the position of a point relative to the origin.
Parallel Vectors: Vectors that have the same direction, regardless of magnitude.
Vector Multiplication: Scalar multiplication affects the magnitude and possibly the direction of a vector.

Important Formulas

Unit Vector:
$\hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|}$
Where $\mathbf{v}$ is a vector and $|\mathbf{v}|$ is its magnitude.
Position Vector:
For a point $P(x, y, z)$ , the position vector from the origin is:
$\mathbf{OP} = x\hat{i} + y\hat{j} + z\hat{k}$
Multiplying a Vector by a Scalar:
$\mathbf{v}' = c \mathbf{v}, \quad \text{where } c \text{ is a scalar.}$
Ratio Formula:
$\underline{r}= \frac{p\mathbf{b}+q\mathbf{a}}{p+q}$

Tips and Tricks

When calculating vector magnitudes, use the Pythagorean theorem:
$|\mathbf{v}| = \sqrt{x^2 + y^2}$
The unit vector is simply the normalized version of any vector.
Remember that for parallel vectors, one can be written as a scalar multiple of the other.

Summary

This exercise covers key vector operations such as addition, subtraction, scalar multiplication, and magnitude calculation. Understanding how vectors are used to solve geometric problems like finding parallel lines and midpoints in shapes is central to mastering vector-based geometry.

Reference

By Sir Shahzad Sair: