Where Mindful Learning Begins
Where Mindful Learning Begins
This Class 12 Maths NCERT Solutions Chapter 10 Ex 10.1 page covers all 5 questions, solved step-by-step, on the two ideas every later exercise in this chapter rests on: telling a scalar (magnitude only) apart from a vector (magnitude and direction), and recognising the different types of vectors — coinitial, equal, collinear and negative — directly from a figure.
Question 1 asks you to represent a displacement graphically, which is really about fixing a scale and a direction correctly — a skill examiners check before anything else in this chapter. Questions 2 and 3 are pure classification questions: for each physical quantity you decide whether direction is meaningful for it or not. Question 4 is the one most students find tricky first time round — reading a square's four side-vectors off a diagram and correctly identifying which pairs are coinitial, which are truly equal (same magnitude and direction, not just same length), and which are collinear but point opposite ways. Question 5 closes the exercise with four true/false statements that examiners love recycling in 1-mark and assertion-reason questions, so it's worth understanding the reasoning behind each one rather than memorising the answer.
Choose a convenient scale, say 1\text{ unit} = 10\text{ km}, so that the displacement of 40 km is represented by a line segment of 4 units.
Draw a ray from the point of origin O towards the north direction (vertically upward). Since the displacement is 30° east of north, rotate this ray by 30° towards the east (i.e., towards the right, clockwise from the north line) and mark the point P at a distance of 4 units along this new direction.
The vector \vec{OP}, drawn from O making an angle of 30° with the north line on the east side, and of length 4 units on the chosen scale, represents the required displacement.
A quantity is a vector only if a direction is meaningful for it; otherwise, it is a scalar.
In the figure, \vec{a} (top side) points rightward, \vec{b} (right side) points downward, \vec{c} (bottom side) points leftward, and \vec{d} (left side) points downward.
(i) Coinitial vectors (same starting point): \vec{a} and \vec{d} both start from the top-left corner of the square, so they are coinitial.
(ii) Equal vectors (same magnitude and same direction): \vec{b} and \vec{d} both point downward and both have length equal to the side of the square, so \vec{b} = \vec{d}.
(iii) Collinear but not equal: \vec{a} and \vec{c} are both parallel to the same (horizontal) pair of lines, so they are collinear; but \vec{a} points rightward while \vec{c} points leftward, so their directions are opposite and they are not equal.
(i) True. Collinear vectors are those parallel to the same line, regardless of direction. Since -\vec{a} is parallel to the line of \vec{a} (just reversed in direction), \vec{a} and -\vec{a} are collinear.
(ii) False. Collinearity depends only on being parallel to a common line; two collinear vectors can have completely different magnitudes.
(iii) False. Two vectors can have equal magnitude while pointing along entirely different (non-parallel) lines, so equal magnitude does not force collinearity.
(iv) False. Two collinear vectors of the same magnitude may still point in opposite directions (like \vec{a} and -\vec{a}), in which case they are not equal — equal vectors need both the same magnitude and the same direction.
Expert CBSE Coaching · Class 9–12