GSEB Class 12 Maths Solutions Chapter 10 Vector Algebra Exercise 10.1

Question 1. Represent graphically a displacement of 40 km, 30° east of north.
Answer: To show this displacement, we draw a line segment 2 cm long to the right of the OY axis. This segment forms an angle of 30° with the OY axis. We set a scale where 1 cm represents 20 km, so 2 cm stands for 40 km. The vector \( \overline{\mathrm{OP}} \) then accurately shows a displacement of 40 km, 30° east of north, as specified.

In simple words: Draw a line starting from the center, pointing towards the top-right corner. Make sure it's 30 degrees away from the up-pointing line (North) and represents 40 km.

Exam Tip: When drawing vectors, always include the origin (O), the vector endpoint (P), the direction (angle with a reference axis), and the magnitude (with a clear scale).

Question 2. Classify the following measures as scalars and vectors:
(i) 10 kg
(ii) 2 metres north-west
(iii) 40°
(iv) 40 watt
(v) \( 10^{-10} \) columb
(vi) 20 m/sec\(^{2}\)
Answer:
Scalars:
(i) 10 kg (This shows only magnitude, like mass.)
(iii) 40° (This denotes an angle, which is a magnitude.)
(iv) 40 watt (This represents power, which has only size.)
(v) \( 10^{-10} \) columb (This is a quantity of electric charge, which is a scalar.)
Vectors:
(ii) 2 metres north-west (This indicates both magnitude (2 metres) and direction (north-west).)
(vi) 20 m/sec\(^{2}\) (This is acceleration, possessing both magnitude and a specific direction.)
In simple words: Scalars are quantities that only tell you "how much," like weight or temperature. Vectors tell you "how much" and "which way," like how fast you're going and in what direction.

Exam Tip: Remember that speed is a scalar, but velocity is a vector. Similarly, distance is a scalar, while displacement is a vector. The key difference is the presence of direction.

Question 3. Classify the following as scalar and vector quantities:
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work
Answer:
Scalar Quantity:
(i) time period (This indicates a duration, which only has magnitude.)
(ii) distance (This measures length, without any specific direction.)
(v) work (This is a form of energy, which is a scalar quantity.)
Vector Quantity:
(iii) force (This is a push or pull, having both magnitude and direction.)
(iv) velocity (This refers to speed in a particular direction.)
In simple words: Some things only have a size (scalars), like how long something takes. Other things have both a size and a direction (vectors), like how hard you push something and in what way.

Exam Tip: To distinguish between scalars and vectors, always ask yourself if a direction is essential to fully describe the quantity. If yes, it's a vector; if no, it's a scalar.

Question 4. Identify the following vectors:
(i) co-initial
(ii) equal
(iii) collinear but not equal
Answer:
(i) Coinitial vectors are \( \vec{a} \) and \( \vec{d} \). (These vectors start from the same initial point in the given diagram.)
(ii) Equal vectors are \( \vec{b} \) and \( \vec{d} \). (These vectors have the same magnitude and direction, making them identical.)
(iii) Collinear but not equal vectors are \( \vec{a} \) and \( \vec{c} \). (These vectors lie on the same line or parallel lines, but their magnitudes or directions are different.)

In simple words: Coinitial vectors start from the same point. Equal vectors have the same length and point in the same direction. Collinear vectors lie on the same line or parallel lines, even if they're not the same length or pointing the same way.

Exam Tip: Always pay close attention to the starting point, magnitude (length), and direction of each vector in the diagram to correctly identify their relationships.

Question 5. Answer the following as true and false:
(i) \( \vec{a} \) and \( - \vec{a} \) are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having the same magnitude are collinear.
(iv) Two collinear vectors having same magnitude are equal.
Answer:
(i) True (A vector \( \vec{a} \) and its negative \( - \vec{a} \) lie on the same line but point in opposite directions, making them collinear.)
(ii) False (Collinear vectors only need to be parallel; their lengths (magnitudes) can be different. For example, \( \vec{a} \) and \( 2\vec{a} \) are collinear but not equal in magnitude.)
(iii) False (Vectors with the same magnitude can point in entirely different directions, meaning they are not necessarily parallel or on the same line, hence not collinear. For example, two vectors representing a 5 km displacement, one towards north and other towards east.)
(iv) False (Collinear vectors with the same magnitude can still be unequal if they point in opposite directions. For instance, \( \vec{a} \) and \( - \vec{a} \) have the same magnitude but are not equal because their directions differ.)
In simple words: "Collinear" means they are on the same line or parallel lines. For vectors, having the same magnitude just means they are the same length. Even if they are on the same line and the same length, they might still be pointing in opposite ways, which means they are not "equal."

Exam Tip: Understand the precise definitions: 'collinear' means parallel, 'equal' means same magnitude AND same direction, and 'same magnitude' only refers to length. Avoid confusing these terms.