VECTORS : CHAPTER # 05 Physics 10th - Question Answers

 Physics 10th - Question Answers

CHAPTER # 05: VECTORS

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Q.1: Define scalar and vector quantities and give five examples of each?

Ans:

Scalar Quantities: Physical quantities which have magnitude and units but there is no need of direction to describe them are called scalar quantities. Scalar quantities can be solved by simple arithmetic rules. They can add, subtract, multiply, and divide easily.

Examples: Mass, length, time, speed, distance, volume, temperature, heat, energy, pressure, power, etc.

Vector Quantities: Physical quantities which have magnitude and units but we need particular direction to completely specify them are called vector quantities. Vectors are added, subtracted, multiplied, and divided by the rules of vector algebra.

Examples: Velocity, displacement, acceleration, force, weight, torque or moment, momentum, etc.

Q.2: How do we represent vector quantities?

Ans: Vector Representation: We can represent vector quantities by an arrow. The length of the arrow represents the magnitude, and the arrowhead represents the direction.

Diagram: A displacement of 50 kilometers due north is represented by the directed line segment $\overrightarrow{AB}$, where:

• Scale: 1 cm = 10 km
• Length: 5 cm

Q.3: Define multiplication of a vector by a number?

Ans: Multiplication of a Vector: When a vector is multiplied by a number, it remains a vector quantity. If the number, say “n,” is positive, the new vector has a magnitude “n” times the magnitude of the original vector, and its direction remains the same.

A line segment making an angle of 30° with the x-axis gives magnitude and direction of velocity of the boat by considering the x-axis parallel to the bank of the river.

Example: For example, a vector $\vec{F}$ of 2 cm length will become of 4 cm if multiplied by 2, and a vector $\vec{F}$ of $\frac{1}{2}$ cm multiplied by 2 will be 1 cm. When a vector is multiplied by a negative number say -2, the new vector becomes two times the magnitude of the original vector, but its direction is opposite to it.

Q.4: Define negative of a vector?

Ans: If we have two vectors of the same magnitude but one of them has the opposite direction to the first vector, it will be the negative of the first vector. The representative lines of a vector and its negative vector are equal and parallel to each other, but their heads are in opposite directions.

Diagram:

• Vector A: 4 cm in the positive direction
• Vector B: 4 cm in the negative direction

The diagram shows two vectors on the coordinate plane where the negative vector is opposite to the positive vector.

Q.5: What do you mean by resultant vector?

Ans:
The process of combining two or more than two vectors to produce a single vector having the combined effect of all the vectors is called the resultant vector.

$$\vec{R}=\vec{A}+\vec{B}$$

Q.6: Explain the addition of vectors by the head-to-tail rule?

Ans:
Two or more vectors can be added by the head-to-tail rule in such a way that we put the tail of the second vector to the head of the first vector and the tail of the third vector to the head of the second vector, and so on. This method of adding vectors is known as the head-to-tail rule of vector addition. There are two methods of adding vectors by the head-to-tail rule.

Diagram:
Illustration of head-to-tail rule of vector addition.

Vector Acting Along the Same Line:
The resultant of a number of vectors that are acting along the same line is a new vector whose magnitude is the sum of the magnitude of all the given vectors and whose direction is the same as that of the given vector.

Vectors Acting Along Different Lines:
The resultant of two vectors acting along different lines can be obtained by drawing the vector lines in such a way that the tail of one coincides with the head of the other. The line joining their free ends will give the resultant vector.

Q.7: Define subtraction of a vector? Explain it with one example?

Ans:
In order to subtract a vector from another vector, the sign of the vector to be subtracted is changed and then added to the other vector.

Example:
If a vector $\vec{B}$ is to be subtracted from a vector $\vec{A}$ then $\vec{A}-\vec{B}$ is found by adding $\vec{A}$ and $-\vec{B}$. The subtraction of vectors is shown in the given figure. Vectors $\vec{A}$, $\vec{B}$, and $-\vec{B}$ are represented by the lines.

$$\vec{R}=\vec{A}+(-\vec{B})=\vec{A}-\vec{B}$$

Figures:

• (a) Two vectors $\vec{A}$ and $\vec{B}$
• (b) Vector $-\vec{B}$
• (c) The addition of $\vec{A}$ and $-\vec{B}$, and the resultant $\vec{R}=\vec{A}-\vec{B}$

Q.8: Define trigonometric ratios?

Ans:
Trigonometry is an important branch of mathematics. It deals with the relations between angles and sides of triangles. A right triangle is one that contains a $90^{\circ }$ angle. Here $AB$ and $BC$ are the adjacent and opposite sides to the angle $\theta$ and are generally called base and perpendicular respectively. The side opposite to the right angle is called hypotenuse. The three most important trigonometric functions of an angle are called sine, cosine, and tangent. They are briefly written as sin, cos, and tan.

The trigonometric ratios give the ratios between the various sides of a right triangle. Here side $AB$ is base or side adjacent to $\theta$ and side $AC$ is hypotenuse which is always opposite the right angle in a right triangle. The common ratios are:

• Sine:

$$\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{P}{H}=\frac{BC}{AC}$$

• Cosine:

$$\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}=\frac{B}{H}=\frac{AB}{AC}$$

Tangent:

$$\text{tan}\theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{P}{B}=\frac{BC}{AB}$$

The inverse functions of sin $\theta$, cos $\theta$, and tan $\theta$ are:

• Cosecant:

$$\text{cosec}\theta =\frac{\text{Hypotenuse}}{\text{Perpendicular}}=\frac{H}{P}=\frac{AC}{BC}$$

• Secant:

$$\text{sec}\theta =\frac{\text{Hypotenuse}}{\text{Base}}=\frac{H}{B}=\frac{AC}{AB}$$

• Cotangent:

$$\text{cot}\theta =\frac{\text{Base}}{\text{Perpendicular}}=\frac{B}{P}=\frac{AB}{BC}$$

Q.9: Make a chart of trigonometric ratios?

Ans: TRIGONOMETRIC RATIOS:

Ratios	$0^{\circ }$	$30^{\circ }$	$45^{\circ }$	$60^{\circ }$	$90^{\circ }$
$\sin \theta$	0.0000	$\frac{1}{2}=0.5$	$\frac{1}{\sqrt{2}}=0.707$	$\frac{\sqrt{3}}{2}=0.866$	1.0000
$\cos \theta$	1.0000	$\frac{\sqrt{3}}{2}=0.866$	$\frac{1}{\sqrt{2}}=0.707$	$\frac{1}{2}=0.5$	0.0000
$\tan \theta$	0.0000	$\frac{1}{\sqrt{3}}=0.577$	1.0000	$\sqrt{3}=1.732$	$\mathrm{\infty }$

Q.10: Define resolution of a vector. By using trigonometric ratios find its horizontal and vertical components?

Ans: RESOLUTION OF A VECTOR:

The process of splitting up a single vector into two or more vectors is called the resolution of a vector. A vector is resolved into its rectangular components.

Mathematical Proof:

Suppose a vector ($\vec{F}$) is represented by the line "AO," making an angle $\theta$ with the horizontal axis OX.

• From point "A," draw "AC" perpendicular to OX.
• The line "OA" represents the horizontal component of vector $\vec{F}$, denoted by $F_x$.
• The line "AC" represents the vertical component, denoted by $F_y$.

Thus, the vector $\vec{F}$ is resolved into two rectangular components.

Determining Components:

The components are determined by considering the right-angled triangle AOB.

$$\cos \theta = \frac{mOA}{mOC}$$

Or:

$$mOA = mOC \cos \theta$$

And:

$$F_x = F \cos \theta$$

Similarly:

$$\sin \theta = \frac{mAC}{mOC}$$

Or:

$$mAC = mOC \sin \theta$$

If we want to find the angle or direction of vector $\vec{F}$, then:

$$\tan \theta = \frac{mAC}{mOA}$$

Or:

$$\tan \theta = \frac{F_y}{F_x}$$

Thus:

$$\theta = \tan^{-1} \left(\frac{F_y}{F_x}\right)$$

Ans.

Q.11: What are Rectangular Components of a Vector? How are They Determined?

Ans: Rectangular Components: If the angle between the horizontal and vertical components is a right angle, then these components are called rectangular components.

Determination of a Vector by Its Rectangular Components: Consider rectangular components $F_x$ and $F_y$, represented by directed line segments $OA$ and $OB$, respectively, in both magnitude and direction. Adding these two components by the head-to-tail rule shows that the directed line segment $OC$ represents the magnitude and direction of vector $\vec{F}$.

To derive the expression for the magnitude and direction of $F$ in terms of its rectangular components $F_x$ and $F_y$, consider the right-angled triangle OAC. By using the Pythagorean theorem, we have:

$$OC^2 = OA^2 + AC^2$$

The direction of $\vec{F}$ is determined by finding the angle $\theta$ that the vector $\vec{F}$ makes with the x-axis in the right-angled triangle OAC. We have:

$$\vec{F}^2 = F_x^2 + F_y^2$$

Or:

$$\vec{F} = \sqrt{F_x^2 + F_y^2}$$

To find the direction:

$$\tan \theta = \frac{AC}{OA}$$

Or:

$$\tan \theta = \frac{F_y}{F_x}$$

Therefore:

$$\theta = \tan^{-1} \left(\frac{F_y}{F_x}\right)$$

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