Vectors

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Class 12 Maths Chapter 10
Vectors
Important Questions

Vectors are an essential topic in Class 12 mathematics, particularly in the context of physics and engineering. They provide a way to represent quantities that have both magnitude and direction. Here are CBSE class 12 maths vectors important questions you should be familiar with.Interacting with diverse question styles helps students confront uncertainties, ensuring they’re fully prepared for the upcoming exams. Handling these varied questions doesn’t just enhance confidence but also sharpens their problem-solving abilities.

Table of Contents

Introduction

Vectors are a fundamental concept in mathematics and physics, essential for understanding various phenomena in multiple dimensions. They are particularly important in fields such as engineering, physics, and computer graphics. Vectors follow certain rules of addition, subtraction, and multiplication (both dot and cross products). These operations obey the associative, distributive, and commutative properties, albeit differently for each type of product.Class 12 vectors important questions and answers help students in understanding the chapter in more detailed way.

What are Vectors?

Vectors are fundamental mathematical objects used to represent quantities that have both magnitude and direction. They differ from scalars, which have only magnitude.
Magnitude and Direction: A vector is characterized by not just how much (magnitude) but also in which direction. For example, in physics, velocity is a vector because it describes both how fast an object is moving (speed) and in which direction.
Graphical Representation:

Vectors are often depicted as arrows. The length of the arrow represents the vector’s magnitude, and the arrow’s direction shows its direction.
The starting point of a vector is called the “tail,” and the end point is the “head.”

Class 12 Vectors Important Questions and Answers

Q1. The position vector of the point (1, 2, 0) is:

Options

(a) $\hat{i} + \hat{j} + \hat{k}\\[4.5 bp]$ (b) $\hat{i} + 2\hat{j} + \hat{k}\\[4.5 bp]$ (c) $\hat{i} + \hat{2j}\\[4.5 bp]$ (d) $2\hat{j} + \hat{k}$

Ans. (c) $\hat{i} + \hat{2j}\\[4.5 bp]$

Q2: The magnitude of the vector

6\hat{i} + 2\hat{j} + 3\hat{k}

is equal to:

Options:

(a) 5
(b) 1
(c) 7
(d) 12

Ans. (c) 7
Explanation:
Vector, V $\rightarrow \space 6\hat{i} + 2\hat{j} + 3\hat{k} \\$ Magnitude of the vector, V;
$\\|V| = \sqrt{(6^2 + 2^2 + 3^2)} = \sqrt{(36+4+9)} = \sqrt{49} = 7$

Q3. Can two different vectors have the same magnitude?

Explanation:
Two vectors can have the same magnitude.
Magnitude of the vector $\hat{i} - 2\hat{j} + \hat{k}$ is equal to the magnitude of the vector
$2\hat{i} + \hat{j} - \hat{k}.$

Q4. The scalar product of

$5\hat{i} + \hat{j} - 3\hat{k}$ and $3\hat{i} - 4\hat{j} +7 \hat{k}$ is ?

Explanation:
A = $5\hat{i} + \hat{j} - 3\hat{k} \\[2.5 bp]$ B = $3\hat{i} - 4\hat{j} + 7\hat{k} \\[2.5 bp]$ A . B = $(5\hat{i} + \hat{j} - 3\hat{k}).(3\hat{i} - 4\hat{j} + 7\hat{k})\\[4.5 bp]$ = 5 · 3 + 1 · (-4) + (-3) · 7
= 15 – 4 – 21
= -10

Q5: Find the projection of the vector

$\hat{i} - \hat{j}$ on the vector $\hat{i} + \hat{j}$

Explanation:
Projection of $\hat{i} - \hat{j}$ on the vector $\hat{i} + \hat{j} \\[4.5 bp] (\hat{i} - \hat{j}).\dfrac{(\hat{i} + \hat{j})}{|\hat{i} + \hat{j}|} = 0$

CBSE Class 12 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Relations and Functions
Chapter 2	Inverse Trigonometric Functions
Chapter 3	Matrices
Chapter 4	Determinants
Chapter 5	Continuity and Differentiability
Chapter 6	Applications of Derivatives
Chapter 7	Integrals
Chapter 8	Applications of the Integrals
Chapter 9	Differential Equations
Chapter 10	Vectors
Chapter 11	Three - dimensional Geometry
Chapter 12	Linear Programming
Chapter 13	Probability

Conclusion

Vectors are a pivotal concept in mathematics and physics, offering a comprehensive way to describe quantities that possess both magnitude and direction. The study of vectors bridges the gap between abstract mathematical theory and real-world applications, making them indispensable in various scientific and engineering fields. oswal.io provide class 12 vectors important questions and answers. Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations, inviting them to delve deeper into the captivating realm of vectors.

Frequently Asked Questions

Q1 : What is a vector?

Ans: A vector is a mathematical object that has both magnitude (size) and direction. It is used to represent quantities that need both of these elements to be fully described, such as force, velocity, or displacement in physics.

Q2 : How is a vector different from a scalar?

Ans: A scalar is a quantity that has only magnitude, like temperature or mass, without any associated direction. A vector, on the other hand, includes both magnitude and direction, such as wind speed (magnitude) blowing towards the north (direction).

Q3 : How are vectors represented?

Ans: Vectors are commonly represented as arrows in diagrams, where the length of the arrow indicates the vector’s magnitude and the arrow’s direction shows its direction. Algebraically, vectors can be represented by listing their components, such as.

Q4 : Can vectors be added or subtracted?

Ans: Yes, vectors can be added and subtracted. Vector addition is typically performed using the triangle or parallelogram method, where vectors are placed head-to-tail. Subtraction involves adding the negative of a vector.

Q5: What is the difference between the dot product and the cross product of vectors?

Ans: The dot product (or scalar product) of two vectors results in a scalar and is indicative of the magnitude of one vector in the direction of the other. The cross product (or vector product) results in a vector that is perpendicular to both vectors involved, which represents the area of the parallelogram they span.

Chapter Wise Important Questions for CBSE Board Class 12 Maths
Relations and Functions
Inverse Trigonometric Functions
Matrices
Determinants
Continuity and Differentiability
Applications of Derivatives
Integrals
Applications of the Integrals
Differential Equations
Vectors
Three - dimensional Geometry
Linear Programming
Probability

Class 12 Maths Chapter 10 Vectors Important Questions