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Physics 1 Marks

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1. Define Scalars.

Scalars are physical quantities that have only magnitude and no direction. Examples include mass, length, time, and speed.

2. Define vectors.

Vectors are physical quantities that have both magnitude and direction. Examples include displacement, velocity, force, and momentum.

3. How do you represent a vector graphically?

A vector is represented graphically by a straight line with an arrowhead. The length of the line represents the magnitude of the vector, and the arrowhead indicates its direction.

4. Define unit vector.

A unit vector is a vector with a magnitude of one. Its purpose is to specify a direction in space. It is denoted by a lowercase letter with a 'hat' symbol above it, like a^. The unit vector of a vector A is given by A^=∣A∣A​.

5. Define null vector.

A null vector (or zero vector) is a vector with a magnitude of zero and an arbitrary direction. It is represented by 0.

6. Define negative vector.

A negative vector of a given vector A is a vector that has the same magnitude as A but points in the opposite direction. It is represented by −A.

7. Write the expression for area of triangle in vector notation.

The area of a triangle formed by two vectors, a and b, is given by:

Area = 21​∣a×b∣

8. Write the expression for area of parallelogram in vector notation.

The area of a parallelogram formed by two vectors, a and b, is given by:

Area = ∣a×b∣

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Physics 3 Marks

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1. Define scalars and vectors and give two examples of each.

Scalars are quantities with only magnitude. Examples: mass and time.

Vectors are quantities with both magnitude and direction. Examples: force and velocity.

2. Explain the working of catapult using parallelogram law of vectors.

The parallelogram law of vectors states that if two vectors acting on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram starting from the same point. A catapult uses this principle. The two ropes of the catapult, when pulled back, represent two force vectors. The tension in each rope is a vector. The resultant force that launches the projectile is the diagonal of the parallelogram formed by these two force vectors. The stronger the tension in the ropes (longer vectors), the greater the resultant force (longer diagonal), and the further the projectile will be launched.

3. Define scalar product and mention two examples of it.

The scalar product (or dot product) of two vectors is a scalar quantity equal to the product of their magnitudes and the cosine of the angle between them. For two vectors A and B, the scalar product is A⋅B=∣A∣∣B∣cosθ.

Two examples:

1. Work done by a force: $W=F\cdot d=Fd\cos \theta$.

2. Electric power: $P=E\cdot v$ (where E is electric field and v is velocity).

4. Define scalar product and write any two properties of it.

The scalar product of two vectors is a scalar quantity obtained by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

Two properties:

1. Commutative: The order of multiplication does not matter. $A\cdot B=B\cdot A$.

2. Distributive: It distributes over vector addition. $A\cdot (B+C)=A\cdot B+A\cdot C$.

5. Write any three properties of scalar product.

1. Commutative: $A\cdot B=B\cdot A$.

2. Dot product of two perpendicular vectors is zero: If $A\perp B$, then $A\cdot B=0$.

3. Dot product of two parallel vectors is maximum: If A∣∣B, then $A\cdot B=|A||B|$.

6. Explain work done by a force as an application of dot product.

Work done is defined as the product of the component of force in the direction of displacement and the magnitude of the displacement. Since both force (F) and displacement (d) are vectors, their scalar product gives the scalar quantity, work done (W). The formula is W=F⋅d=∣F∣∣d∣cosθ, where θ is the angle between the force and displacement vectors. This equation shows that only the component of force that is in the direction of motion does work. For example, if a force is applied at a 90∘ angle to the displacement, no work is done.

7. Define vector product of two vectors and write two examples of it.

The vector product (or cross product) of two vectors is a vector quantity with a magnitude equal to the product of their magnitudes and the sine of the angle between them. Its direction is perpendicular to the plane containing the two vectors, determined by the right-hand rule. For two vectors A and B, the vector product is A×B=∣A∣∣B∣sinθn^, where n^ is the unit vector perpendicular to the plane.

Two examples:

1. Torque: $\tau =r\times F$.

2. Angular momentum: $L=r\times p$.

8. Define cross product and write any two properties of it.

The cross product of two vectors A and B is a vector whose magnitude is ∣A∣∣B∣sinθ and direction is perpendicular to the plane containing A and B.

Two properties:

1. Non-commutative: The order of multiplication matters. $A\times B=-(B\times A)$.

2. Cross product of two parallel vectors is a null vector: If A∣∣B, then $A\times B=0$.

9. Write any three properties of vector product.

1. Non-commutative: $A\times B\ne B\times A$. Instead, $A\times B=-(B\times A)$.

2. Distributive: It distributes over vector addition. $A\times (B+C)=(A\times B)+(A\times C)$.

3. Cross product of a vector with itself is a null vector: $A\times A=0$.

10. Explain torque as an application of vector product.

Torque is the rotational equivalent of force. It is the measure of the turning effect of a force on an object. The turning effect depends on both the magnitude of the force and the distance from the pivot point (the position vector r). The relationship is given by the cross product: τ=r×F. This vector equation correctly shows that the torque is a vector perpendicular to both the position vector and the force vector. The magnitude of the torque is ∣r∣∣F∣sinθ, which indicates that maximum torque is produced when the force is applied perpendicular to the position vector (at θ=90∘).

11. Write the expressions for areas of triangle and parallelogram in vector notation.

• Area of a triangle: Area = $\frac{1}{2}|a\times b|$

• Area of a parallelogram: Area = $|a\times b|$

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Physics 5 Marks

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1. State and explain triangle law of vectors.

Triangle Law of Vectors: If two vectors are represented by two sides of a triangle taken in the same order, then their resultant is represented by the third side of the triangle taken in the opposite order.

Explanation: Consider two vectors A and B. To find their resultant R, we place the tail of vector B at the head of vector A. The resultant vector R is then drawn from the tail of A to the head of B. The vector sum is R=A+B. This forms a triangle, where the two vectors are two sides and the resultant is the third side.

2. State and explain parallelogram law of vectors.

Parallelogram Law of Vectors: If two vectors acting at a point are represented by the two adjacent sides of a parallelogram, their resultant is given by the diagonal of the parallelogram drawn from the same point.

Explanation: Consider two vectors A and B acting at a point. We draw them as the adjacent sides of a parallelogram. The resultant vector R is the diagonal of the parallelogram that starts from the same initial point. This law provides a graphical method to find the resultant of two vectors. The vector sum is R=A+B.

3. Derive expressions for magnitude and direction of resultant of two vectors using parallelogram law of vectors.

Let two vectors A and B act at a point, making an angle θ with each other. According to the parallelogram law, the resultant vector R is the diagonal of the parallelogram.

Magnitude of the Resultant:

Using the law of cosines in the triangle formed by the vectors, we get the magnitude of the resultant:

R2=A2+B2+2ABcosθ

R=A2+B2+2ABcosθ​

Direction of the Resultant:

Let the resultant vector R make an angle α with vector A.

Using the law of sines in the triangle:

sinαB​=sin(180∘−θ)R​

sinα=RBsin(180∘−θ)​=RBsinθ​

α=tan−1(A+BcosθBsinθ​)

4. Define scalar product and write any four properties of scalar product.

The scalar product of two vectors A and B is a scalar quantity equal to the product of their magnitudes and the cosine of the angle between them.

Four properties:

1. Commutative: $A\cdot B=B\cdot A$.

2. Distributive: $A\cdot (B+C)=A\cdot B+A\cdot C$.

3. Dot product of a vector with itself: $A\cdot A=|A{|}^2=A^2$.

4. Orthogonality: If two vectors are perpendicular ($\theta =90^{\circ }$), their scalar product is zero. $A\cdot B=0$.

5. Write any five properties of dot product.

1. Commutative: The order of the vectors doesn't change the result. $A\cdot B=B\cdot A$.

2. Distributive: It distributes over vector addition. $A\cdot (B+C)=A\cdot B+A\cdot C$.

3. Orthogonality: The dot product of two perpendicular vectors is zero.

4. Parallel vectors: The dot product of two parallel vectors is the product of their magnitudes. $A\cdot B=|A||B|$.

5. Self-dot product: The dot product of a vector with itself gives the square of its magnitude. $A\cdot A=|A{|}^2$.