Math Case Study Vectors with Answers Class 12 Solved Questions

Chapter: Math Case Study Vectors with Answers Class 12

The chapter on Math Case Study Vectors with Answers Class 12 helps students understand vector algebra using real-life mathematical situations. Moreover, it explains concepts clearly through solved examples. Therefore, students gain confidence as they work through application-based vector questions that improve accuracy and problem-solving skills.

Core Concepts of Vector Algebra

This section introduces dot product, cross product, and vector projections. Additionally, students learn how these operations are applied in different case study problems. These concepts make solving vector-based questions easier and more structured.

Importance of Case Study Practice

Practicing case studies enhances logical reasoning. Consequently, students understand how vectors connect to real applications. This approach not only boosts exam preparation but also strengthens overall mathematical understanding.

Math Case Study Vectors with Answers Class 12 Solved Questions

In a physics laboratory, a team of students is studying the motion of an object on a smooth surface. They record the displacement of the object using directed line segments and understand that these can be represented as **vectors**. The team observes how vectors behave when applied in different directions and magnitudes. They begin by identifying the basic concepts — such as **zero vector**, **unit vector**, and **position vector**. During their observations, they also discuss the concept of **equal** and **opposite vectors** and how two vectors can be **collinear** or **coplanar**. They use arrows on graph paper to draw and represent these vectors and verify the properties of vector addition and scalar multiplication.

Theory and Formulae Related to Vectors:

  • A vector is a quantity that has both **magnitude** and **direction**.
  • A **zero vector** has zero magnitude: $\vec{0} = 0\hat{i} + 0\hat{j} + 0\hat{k}$.
  • A **unit vector** has magnitude 1: $|\vec{a}| = 1$.
  • Two vectors are **equal** if they have the same magnitude and direction.
  • **Collinear vectors**: $\vec{a} = \lambda \vec{b}$ for some scalar $\lambda$.
  • **Position vector** of a point $P(x, y, z)$: $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$.

1. Which of the following is a unit vector?

Solution:
A unit vector must have a magnitude of 1.
For option (b), let $\vec{v} = \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$.
Magnitude $|\vec{v}| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1$.
Correct answer is option **(b)**.

2. If $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$, what is the magnitude of $\vec{a}$?

Solution:
The magnitude of a vector $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$ is $|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$.
Here, $|\vec{a}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29}$.
Correct answer is option **(b)**.

3. Which of the following is a zero vector?

Solution:
A zero vector is a vector whose magnitude is zero. In component form, all components are zero.
$\vec{0} = 0\hat{i} + 0\hat{j} + 0\hat{k}$.
Correct answer is option **(b)**.

4. If $\vec{a} = \hat{i} + 2\hat{j}$ and $\vec{b} = 2\hat{i} + 4\hat{j}$, then which of the following is true?

Solution:
Collinear vectors satisfy the condition $\vec{b} = \lambda \vec{a}$ for some scalar $\lambda$.
$\vec{b} = 2\hat{i} + 4\hat{j} = 2(\hat{i} + 2\hat{j})$. Since $\vec{a} = \hat{i} + 2\hat{j}$, we have $\vec{b} = 2\vec{a}$.
Since $\lambda = 2$ is a scalar, the vectors are collinear.
Correct answer is option **(c)**.

5. Which statement is correct for equal vectors?

Solution:
By definition, two vectors are equal if and only if they have the same magnitude and the same direction.
Correct answer is option **(c)**.

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