Unit Vectors MCQ Quiz - Objective Question with Answer for Unit Vectors - Download Free PDF

Given:

\(\vec{a}=\hat{i}+\hat{j}+\hat{k}\)

\(\vec{b}=2\hat{i}-\hat{j}+3\hat{k}\)

\(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\)

Calculation:

The unit vector in the direction of the vector,  \(\vec r = 2\vec{a}-\vec{b}+3\vec{c}\)

⇒ \(\vec r = 2(\hat i + \hat j + \hat k) - (2\hat i - \hat j + 3\hat k) + 3(\hat i - 2\hat j + \hat k)\)

⇒ \(\vec r = 2 \hat i + 2\hat j + 2\hat k - 2 \hat i + \hat j - 3\hat k + 3 \hat i - 6 \hat j + 3 \hat k\)

⇒ \(\vec r = (2 - 2 + 3) \hat i + (2 + 1 - 6) \hat j + (2 - 3 + 3) \hat k\)

⇒ \(\vec r = 3 \hat i - 3 \hat j + 2 \hat k\)

Magnitude or \(\vec r\) = \(\sqrt{3^2 + (-3)^2 + 2^2}\)

⇒ \(|\vec r|\) = \(\sqrt{9+ 9+ 4}\) = \(\sqrt{22}\)

Unit vector in the direction of \(\vec r = \frac{1}{| \vec r|}\times \vec r\)

⇒ \(\vec r = \frac{1}{\sqrt{22}} \times [3 \hat i - 3 \hat j + 2 \hat k]\)

⇒ \(\vec r = \frac{3}{\sqrt{22}} \hat i - \frac{3}{\sqrt{22}} \hat j + \frac{2}{\sqrt{22}} \hat k\)

∴ The required vector is \(\frac{3}{\sqrt{22}} \hat i - \frac{3}{\sqrt{22}} \hat j + \frac{2}{\sqrt{22}} \hat k\)

The correct answer is Vector.

Key Points

• Momentum is a vector quantity, which means it has both magnitude and direction.
• It is defined as the product of an object's mass and its velocity.
• The SI unit of momentum is kilogram meter per second (kg·m/s).
• In physics, momentum is a fundamental concept for understanding motion and the effects of forces on objects.

Important Points

• Momentum is conserved in isolated systems, which means the total momentum of the system remains constant if no external forces are acting on it.
• Impulse is related to momentum, where impulse is the change in momentum resulting from a force applied over a period of time.

Additional Information

• Tensor: A tensor is a mathematical object that generalizes scalars, vectors, and matrices. Tensors are used in various fields including physics and engineering for representing complex relationships.
• Scalar: A scalar is a quantity that has only magnitude and no direction. Examples include mass, temperature, and energy.
• Matrix: A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are used in various branches of mathematics and engineering for solving systems of linear equations and other applications.

The correct answer is Magnetisation.

Key Points

• Represents the magnetic moment per unit volume of a material.
• Describes the extent to which a material becomes magnetized in the presence of an external magnetic field.
• It is a vector quantity because it has both magnitude and direction.
• The direction of magnetisation indicates the alignment of magnetic moments within the material.

Additional Information

Explanation:

Zero vector/Null vector:

• The vector having magnitude equal to zero is called a null vector. It is generally represented by O.
• In zero vector the initial and terminal points coincide with each other. 
• A point is generally taken as a null vector.
• \(\left| {\vec A} \right| = 0\)

Free vector:

• If a vector's initial point is not defined then it is said to be a free vector.

Unit vector:

• A vector that has a unit magnitude (length) is called a unit vector.

Equal vectors:

• When the magnitude and direction of two vectors are same, the vector is said to be equal vector.

The correct answer is 100 kilograms.

Key Points

• A quintal is a metric unit of mass, equal to 100 kilograms.
• This unit is commonly used in many countries for measuring agricultural produce like grains and other commodities.
• The term "quintal" is derived from the French word "quintal" which means a hundredweight.
• In some countries, a quintal is also referred to as a "metric quintal" to distinguish it from other similar units of mass.

Important Points

• The quintal is widely used in agriculture and farming sectors for trade and measurement of large quantities of produce.
• It is a part of the metric system, which is used globally for standardization of weights and measures.
• One quintal is equivalent to 100 kilograms, making it easy to convert between these two units.

Explanation:

Zero vector/Null vector:

• The vector having magnitude equal to zero is called a null vector. It is generally represented by O.
• In zero vector the initial and terminal points coincide with each other. 
• A point is generally taken as a null vector.
• \(\left| {\vec A} \right| = 0\)

Free vector:

• If a vector's initial point is not defined then it is said to be a free vector.

Unit vector:

• A vector that has a unit magnitude (length) is called a unit vector.

Equal vectors:

• When the magnitude and direction of two vectors are same, the vector is said to be equal vector.

Concept:

For two vectors \(\rm \vec A\) and \(\rm \vec B\), the resultant will be:

\(\rm \left|\vec A+\vec B\right|^2=\left|\vec A\right|^2+\left|\vec B\right|^2 + 2\left(\vec A.\vec B\right)\;\)

Calculation:

Given:

\(|\vec A|=|\vec B|=1=|\vec A + \vec B|\)

\(\rm \left|\vec A+\vec B\right|^2=\left|\vec A\right|^2+\left|\vec B\right|^2 + 2\left(\vec A.\vec B\right)\;\)

\(1=1+1 + 2|\vec A|.|\vec B|\cos θ\;\)

\(1=1+1 + (2\times 1 \times 1)\cos θ\;\)

cos θ = -1/2

\(\therefore \theta =\frac{2 \pi}{3}=120^{\circ}\)

CONCEPT:

• The magnitude of a vector: The magnitude of a vector \(\overrightarrow{A}=a\widehat{i}+b\widehat{j}+c\widehat{k}\) is given by:

\(|\overrightarrow{A}|=√ {a^2+b^2+c^2}\)

• Unit vector: The vector that has a magnitude of one unit is known as a unit vector.

EXPLANATION:

Given that \(\overrightarrow{A}=x\widehat{i}+0.5\widehat{j}-0.2\widehat{k}\)

Since this is a unit vector, So its magnitude will be 1.

\(|\overrightarrow{A}|=1\)

Magnitude fo vector A 

\(|\overrightarrow{A}|=√ {x^2+0.5^2+(-0.2)^2}\)

\(1=√ {x^2+0.5^2+(-0.2)^2}\)

​\(1= {x^2+0.5^2+(-0.2)^2}\)​​

\(1={x^2+0.25+0.04}\)​

x = √0.71

So the correct answer is option 4.

The correct answer is option 3) i.e. \(\frac{\hat i}{\sqrt2} + \frac{\hat j}{\sqrt2}\)

CONCEPT:​

• Vector quantities: Vector quantities are those quantities that have two characteristics - a magnitude and direction.
• 2D vectors are represented using the two co-ordinate points in the form: xî + yĵ 

The magnitude of this vector is calculated as follows: \(\sqrt{{x^2 + y^2}}\)  

• 3D vectors are represented using the three co-ordinate points in the form: xî + yĵ + zk̂ 

The magnitude of this vector is calculated as follows: \(\sqrt{x^2 + y^2 + z^2}\)

• Unit vector: A vector that has a magnitude of 1 is called a unit vector.

​ \(\hat i + \hat j ​​\Rightarrow \sqrt{{1^2 + 1^2}} = \sqrt2\)

\(\hat i + \frac{\hat j}{\sqrt2} \Rightarrow \sqrt{1^2 + (\frac{1}{\sqrt2})^2} =\sqrt{\frac{3}{2}}\)

\(\frac{\hat i}{\sqrt2} + \frac{\hat j}{\sqrt2} \Rightarrow \sqrt{(\frac{1}{\sqrt2})^2 + (\frac{1}{\sqrt2})^2} = 1\)

Thus, \(\frac{\hat i}{\sqrt2} + \frac{\hat j}{\sqrt2}\) is the unit vector.

Given:

Vector a = î + ĵ + k̂

Vector b = 2î - ĵ + 3k̂

Vector c = î - 2ĵ + k̂

Calculation:

The unit vector in the direction of the vector, r = 2a - b + 3c

⇒ r = 2(î + ĵ + k̂) - (2î - ĵ + 3k̂) + 3(î - 2ĵ + k̂)

⇒ r = 2î + 2ĵ + 2k̂ - 2î + ĵ - 3k̂ + 3î - 6ĵ + 3k̂

⇒ r = (2 - 2 + 3)î + (2 + 1 - 6)ĵ + (2 - 3 + 3)k̂

⇒ r = 3î - 3ĵ + 2k̂

Magnitude of r = √(3² + (-3)² + 2²)

⇒ |r| = √(9 + 9 + 4) = √22

Unit vector in the direction of r = (1 / |r|) × r

⇒ Unit vector = (1 / √22) × (3î - 3ĵ + 2k̂)

⇒ Unit vector = (3 / √22)î - (3 / √22)ĵ + (2 / √22)k̂

∴ The required vector is (3 / √22)î - (3 / √22)ĵ + (2 / √22)k̂

Given:

Surface is x2 + y2 – z2 = 11

At point (4, 2, 3)

Formula used:

Unit vector perpendicular to the surface at point 'n' given by \(\frac{\hat{n}}{\hat{|n|}}\)

Calculation:

\({\hat{|n|}} = \sqrt{4^2 + 2^2 + (-3)^2} = \sqrt{29}\)

∴ The unit vector perpendicular to the surface x2 + y2 – z2 = 11 at the point (4, 2, 3) = \(\frac{{4\widehat i + 2\widehat j - 3\widehat k}}{{\sqrt {29} }}\)

The correct answer is Vector.

Key Points

• Momentum is a vector quantity, which means it has both magnitude and direction.
• It is defined as the product of an object's mass and its velocity.
• The SI unit of momentum is kilogram meter per second (kg·m/s).
• In physics, momentum is a fundamental concept for understanding motion and the effects of forces on objects.

Important Points

• Momentum is conserved in isolated systems, which means the total momentum of the system remains constant if no external forces are acting on it.
• Impulse is related to momentum, where impulse is the change in momentum resulting from a force applied over a period of time.

Additional Information

• Tensor: A tensor is a mathematical object that generalizes scalars, vectors, and matrices. Tensors are used in various fields including physics and engineering for representing complex relationships.
• Scalar: A scalar is a quantity that has only magnitude and no direction. Examples include mass, temperature, and energy.
• Matrix: A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are used in various branches of mathematics and engineering for solving systems of linear equations and other applications.

Explanation:

Zero vector/Null vector:

• The vector having magnitude equal to zero is called a null vector. It is generally represented by O.
• In zero vector the initial and terminal points coincide with each other. 
• A point is generally taken as a null vector.
• \(\left| {\vec A} \right| = 0\)

Free vector:

• If a vector's initial point is not defined then it is said to be a free vector.

Unit vector:

• A vector that has a unit magnitude (length) is called a unit vector.

Equal vectors:

• When the magnitude and direction of two vectors are same, the vector is said to be equal vector.

Concept:

For two vectors \(\rm \vec A\) and \(\rm \vec B\), the resultant will be:

\(\rm \left|\vec A+\vec B\right|^2=\left|\vec A\right|^2+\left|\vec B\right|^2 + 2\left(\vec A.\vec B\right)\;\)

Calculation:

Given:

\(|\vec A|=|\vec B|=1=|\vec A + \vec B|\)

\(\rm \left|\vec A+\vec B\right|^2=\left|\vec A\right|^2+\left|\vec B\right|^2 + 2\left(\vec A.\vec B\right)\;\)

\(1=1+1 + 2|\vec A|.|\vec B|\cos θ\;\)

\(1=1+1 + (2\times 1 \times 1)\cos θ\;\)

cos θ = -1/2

\(\therefore \theta =\frac{2 \pi}{3}=120^{\circ}\)

CONCEPT:

• The magnitude of a vector: The magnitude of a vector \(\overrightarrow{A}=a\widehat{i}+b\widehat{j}+c\widehat{k}\) is given by:

\(|\overrightarrow{A}|=√ {a^2+b^2+c^2}\)

• Unit vector: The vector that has a magnitude of one unit is known as a unit vector.

EXPLANATION:

Given that \(\overrightarrow{A}=x\widehat{i}+0.5\widehat{j}-0.2\widehat{k}\)

Since this is a unit vector, So its magnitude will be 1.

\(|\overrightarrow{A}|=1\)

Magnitude fo vector A 

\(|\overrightarrow{A}|=√ {x^2+0.5^2+(-0.2)^2}\)

\(1=√ {x^2+0.5^2+(-0.2)^2}\)

​\(1= {x^2+0.5^2+(-0.2)^2}\)​​

\(1={x^2+0.25+0.04}\)​

x = √0.71

So the correct answer is option 4.