Vector Algebra Ex 10.1 Class 12 NCERT: Concepts and Solutions

Scalars and Vectors: The Core Distinction

In our study of mathematics and physics, we encounter different kinds of physical quantities. The first step in vector algebra is to classify them.

Scalars: A quantity that is defined by its magnitude (or size) alone is called a scalar. It has no direction associated with it. For example, if we say the temperature is 30°C, or the mass of a block is 5 kg, we have completely described these quantities. There's no direction involved. Other examples include length, time, speed, volume, and density.

Vectors: A quantity that requires both magnitude and a specific direction to be fully described is called a vector. Force is a classic example. Just saying you applied a force of 10 Newtons isn't enough; you must also specify the direction (e.g., downwards, or 30° to the horizontal). Other examples are displacement, velocity, and acceleration. We represent a vector graphically as a directed line segment. The length of the segment corresponds to the vector's magnitude, and the arrowhead indicates its direction. We denote a vector from point A (initial) to point B (terminal) as vec(AB) or with a single letter like vec(a). Its magnitude is written as |vec(AB)| or |vec(a)|.

Key Vector Types in Ex 10.1

Zero Vector (Null Vector)

A vector whose initial and terminal points coincide. It has a magnitude of zero and its direction is not defined (or indeterminate). It is denoted by vec(0).

Unit Vector

A vector with a magnitude of exactly one unit. It is used to specify a particular direction. The unit vector in the direction of a vector vec(a) is denoted by â and is calculated as â = vec(a) / |vec(a)|.

Coinitial Vectors

Two or more vectors which have the same starting (initial) point are called coinitial vectors.

Collinear Vectors

Two or more vectors are said to be collinear if they are parallel to the same line, regardless of their magnitudes and directions. For any two collinear vectors vec(a) and vec(b), one can be written as a scalar multiple of the other, i.e., vec(a) = λ * vec(b).

Equal Vectors

Two vectors vec(a) and vec(b) are considered equal if they have the same magnitude and the same direction, regardless of the positions of their initial points. We write this as vec(a) = vec(b).

Worked Examples on Vector Basics

• Problem 1: Represent graphically a displacement of 40 km, 30° east of north. Solution: Step 1: Draw a coordinate system with the vertical axis representing North-South and the horizontal axis representing East-West. Step 2: The direction is '30° east of north'. This means you start by facing North and then turn 30° towards the East. Step 3: Draw a line from the origin at this 30° angle into the first quadrant. Step 4: Choose a suitable scale, for instance, 1 cm = 10 km. The length of your line segment should be 40/10 = 4 cm. * Step 5: Draw a 4 cm long line segment from the origin along the determined direction and place an arrowhead at the end. This arrow represents the displacement vector.
• Problem 2: Classify the following measures as scalars and vectors: (i) 15 kg (ii) 2 meters north-west (iii) 45° (iv) 100 watts (v) 20 m/s² Solution: Step 1: Analyze each quantity for the presence of magnitude and direction. (i) 15 kg: This is mass. It has only magnitude. Scalar. (ii) 2 meters north-west: This is displacement. It has magnitude (2 meters) and direction (north-west). Vector. (iii) 45°: This is an angle. It has only magnitude. Scalar. (iv) 100 watts: This is power. It has only magnitude. Scalar. (v) 20 m/s²: This is acceleration. It has both magnitude and direction (even if not explicitly stated, acceleration is inherently directional). Vector.

Exam Tip: Distinguishing Collinear vs. Equal Vectors

A common trap in exams is confusing 'collinear' and 'equal' vectors. Remember this key difference:

• Collinear Vectors only care about being parallel. They can have different magnitudes and even point in opposite directions. For example, a vector vec(a) and another vector -2vec(a) are collinear because they lie on parallel lines, but they are clearly not equal.

• Equal Vectors are much more restrictive. They must have the same magnitude AND the same direction. They are essentially identical copies of each other, possibly shifted in space.

Exam Tactic: If a question asks if vectors are collinear, check if one is a scalar multiple of the other (vec(b) = λ vec(a)). If it asks if they are equal, you must verify that λ=1 specifically, meaning they are identical in both length and direction.

Practice Questions with Solutions

• Q: Represent graphically a force of 100 N applied in the direction 60° south of west. A: Step 1: Draw the North-South and East-West axes. The 'West' direction is your primary reference. Step 2: From the West direction, measure an angle of 60° towards the South. This will be in the third quadrant. Step 3: Choose a scale, for example, 1 cm = 20 N. The required length of the vector will be 100 N / 20 N/cm = 5 cm. Step 4: Draw a 5 cm line segment from the origin at the calculated angle and add an arrowhead at the terminal point to represent the force vector. Final answer: A 5 cm line segment pointing into the third quadrant, at an angle of 60° below the negative x-axis (West).
• Q: In the given figure, a square ABCD is shown. Identify the following: (i) Equal vectors (ii) Collinear but not equal vectors. [Imagine a square ABCD in order, with vector arrows on all sides: A->B, B->C, C->D, D->A] A: Step 1: Analyze the properties of a square. All sides have equal length, and opposite sides are parallel. Step 2: For (i) Equal vectors, we need vectors with the same magnitude and same direction. vec(AB) and vec(DC) are equal in magnitude and parallel. However, the vector from D to C (vec(DC)) has the same direction as vec(AB). So, vec(AB) = vec(DC). Similarly, vec(DA) and vec(CB) are equal. Step 3: For (ii) Collinear but not equal vectors, we need vectors that are parallel but not identical. vec(AB) and vec(CD) are collinear (parallel) but have opposite directions, so they are not equal. Similarly, vec(BC) and vec(DA) are collinear but not equal. Final answer: (i) Equal vectors: vec(AB) and vec(DC); vec(BC) and vec(AD). (ii) Collinear but not equal: vec(AB) and vec(CD); vec(BC) and vec(DA).
• Q: A man walks 3 km East and then 4 km North. What is the magnitude of his displacement vector? A: Step 1: Let the eastward displacement be vector vec(a) and northward displacement be vector vec(b). The man's path forms a right-angled triangle with the displacement vector as the hypotenuse. Step 2: The magnitude of the eastward displacement is |vec(a)| = 3 km. The magnitude of the northward displacement is |vec(b)| = 4 km. These two vectors are perpendicular. Step 3: The magnitude of the resultant displacement vector vec(r) can be found using the Pythagorean theorem: |vec(r)|² = |vec(a)|² + |vec(b)|². Step 4: Substitute the values: |vec(r)|² = 3² + 4² = 9 + 16 = 25. Step 5: Take the square root: |vec(r)| = sqrt(25) = 5. Final answer: The magnitude of his displacement vector is 5 km.
• Q: Given two vectors vec(p) = î + ĵ + k̂ and vec(q) = -2î - 2ĵ - 2k̂. Are they collinear? A: Step 1: To check for collinearity, we see if one vector is a scalar multiple of the other. That is, can we find a scalar λ such that vec(q) = λ vec(p)? Step 2: Write out the equation using the vector components: -2î - 2ĵ - 2k̂ = λ(î + ĵ + k̂). Step 3: This gives -2î - 2ĵ - 2k̂ = λî + λĵ + λk̂. Step 4: By comparing the coefficients of î, ĵ, and k̂, we get three equations: λ = -2, λ = -2, and λ = -2. Since λ has a consistent value (-2) for all components, the condition is met. Final answer: Yes, the vectors are collinear because vec(q) = -2 vec(p).

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