03_Ex 7.3

Exercise Questions

Questions	Links
Q1. Find the cosine of the angle between $\underline{U}$ and $\underline{V}$…	7.3 Q-1
Q2. Calculate the projection of $\underline{\mathbf{a}}$ along…	7.3 Q-2
Q3. Find a real number $\alpha$ so that $\underline{U}$ and…	7.3 Q-3
Q4. Find the number ” $z$ ” so that the triangle…	7.3 Q-4
Q5. If $V$ is a vector for which…	7.3 Q-5
Q6. Show that the vectors…	7.3 Q-6
Q7. Show that mid-point of hypotenuse of a right angle…	7.3 Q-7
Q8. Prove that the perpendicular bisectors of the…	7.3 Q-8
Q9. Prove that the altitudes of a triangle are…	7.3 Q-9
Q10. Prove that the angle in a semi-circle is a…	7.3 Q-10
Q11. Prove that $\cos (\alpha+\beta)=\cos \alpha \cdot \cos \beta-\sin \alpha \cdot \sin \beta$	7.3 Q-11
Q12. Prove that in any angle $A B C$…	7.3 Q-12

---

Overview

This exercise covers the fundamental concepts of vectors, specifically focusing on the dot product (or scalar product), vector magnitude, projections, and direction cosines. Understanding the dot product is essential in geometry and physics as it allows for calculations involving angles between vectors, projections of vectors, and determining orthogonality (perpendicularity) of vectors.

The exercise will explore key operations on vectors such as:

• Calculating the dot product and its properties.
• Understanding the projection of one vector onto another.
• Working with direction cosines and angles between vectors.

These concepts are applicable in many areas, including mechanics, electromagnetism, and computer graphics.

---

Key Concepts

1. Dot Product (Scalar Product):
   The dot product between two vectors u and v is defined as:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

where $\theta$ is the angle between the vectors.

2. Properties of the Dot Product:
   • Commutative Law: $\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}$
   • Distributive Law: $\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}$
   • Self-Product: $\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2$
   • Orthogonality: If u and v are perpendicular, $\mathbf{u} \cdot \mathbf{v} = 0$.
   • Projection: The projection of one vector onto another is given by:

$$\text{Proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$$

3. Direction Cosines:
   The direction cosines of a vector are the cosines of the angles between the vector and the $x$-, $y$-, and $z$-axes. They are calculated as:

$$\cos \alpha = \frac{v_x}{|\mathbf{v}|}, \quad \cos \beta = \frac{v_y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{v_z}{|\mathbf{v}|}$$

---

Important Formulas

1. Magnitude of a Vector: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

2. Dot Product of Two Vectors: $\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z$

3. Projection of a Vector:

   • Projection of u onto v: $\text{Proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$

4. Cosine of the Angle Between Two Vectors: $\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$

5. Direction Cosines: $\cos \alpha = \frac{v_x}{|\mathbf{v}|}, \quad \cos \beta = \frac{v_y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{v_z}{|\mathbf{v}|}$

---

Tips and Tricks

• Check Perpendicularity: To check if two vectors are perpendicular, calculate their dot product. If the result is zero, the vectors are perpendicular.

• Projection Calculation: When projecting one vector onto another, remember that the projection is a scalar value representing the component of the first vector along the direction of the second vector.

• Right-Angle Condition in Triangles: If a triangle is formed by vectors, check if the sum of the vectors leads to the zero vector, indicating a closed triangle (i.e., the vectors form a closed shape).

• Use Magnitudes and Angles: For problems involving angles between vectors, always start by calculating the magnitudes of the vectors and then use the formula for the cosine of the angle: $\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$

---

Summary

This exercise explores the dot product, a fundamental operation for finding the angle between two vectors, as well as other important concepts such as projections, magnitude, and direction cosines. The dot product is crucial for determining whether two vectors are perpendicular and for projecting vectors onto one another. Understanding these principles allows for deeper insights into vector geometry and is widely applied in physics and engineering fields.

---

Reference

By Sir Shahzad Sair: