Column I gives four relations between the vectors veca, vecb and vecc; Column II gives four orientations of th

NCERT Exemplar · Q26

\vec{a}

(a)

\vec{a} + \vec{b} = \vec{c}

(b)

\vec{a} - \vec{c} = \vec{b}

(c)

\vec{b} - \vec{a} = \vec{c}

(d)

\vec{a} + \vec{b} + \vec{c} = 0

Column II (each in the plane, tail-to-head arrangement):

(i) From the origin,

\vec{a}

points vertically upward; from the tip of

\vec{a}

\vec{c}

points horizontally to the right; and

\vec{b}

runs as the diagonal from the origin up to the tip of

\vec{c}

(up and to the right). Thus

\vec{a}

and

\vec{c}

are placed head-to-tail and

\vec{b}

joins the common tail to the final head.

(ii) From the origin,

\vec{a}

points horizontally to the right; from the tip of

\vec{a}

\vec{b}

points vertically upward; and

\vec{c}

runs from the tip of

\vec{b}

diagonally back down to the origin (down and to the left), closing the triangle so that the three arrows form one continuous loop.

(iii) From the origin,

\vec{a}

points vertically upward; from the origin,

\vec{c}

points horizontally to the left; and

\vec{b}

runs as the diagonal from the tip of

\vec{c}

up to the tip of

\vec{a}

(up and to the right). Thus

\vec{c}

and

\vec{b}

are head-to-tail and

\vec{a}

joins the common tail to the final head.

(iv) From the origin,

\vec{a}

points horizontally to the left; from the tip of

\vec{a}

\vec{b}

points vertically downward; and

\vec{c}

runs as the diagonal from the origin down to the tip of

\vec{b}

(down and to the left). Thus

\vec{a}

and

\vec{b}

are head-to-tail and

\vec{c}

joins the common tail to the final head. Match (a), (b), (c),

(d) to (i)-(iv).

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Concept understanding — Triangle Inequality

Triangle Inequality (Bounding the Resultant of Two Vectors)

When you combine two displacements, two velocities, or two forces in "Motion in a Plane," you add them as vectors using the triangle law: place the tail of the second vector at the head of the first, and the resultant runs from the start to the finish. The triangle inequality is simply the statement that this resultant can never be longer than the two vectors laid end to end, and can never be shorter than their difference.

The Intuition

Suppose you walk 3 m in one direction, then 4 m in some other direction. Could you end up 8 m from where you started? No — the farthest you can possibly get is $3+4=7$ m, and that only happens if both walks point the same way, so there's no bend at all (a "flat" triangle). The moment the second walk points in a different direction, a real corner appears, and cutting across that corner (the direct path) is always shorter than going via the corner. That's the geometric heart of every triangle: any one side is shorter than the sum of the other two, unless the triangle collapses onto a straight line.

The Precise Statement

For two vectors $\vec{A}$ and $\vec{B}$ added by the triangle law, the magnitude of the resultant $\vec{R} = \vec{A} + \vec{B}$ is bounded on both sides:

|\vec{A}| - |\vec{B}|\big| \;\le\; |\vec{A}+\vec{B}| \;\le\; |\vec{A}| + |\vec{B}|

The upper bound $|\vec{A}+\vec{B}| \le |\vec{A}|+|\vec{B}|$ is reached only when $\vec{A}$ and $\vec{B}$ point in exactly the same direction (the angle between them is $0^\circ$ ) — the triangle flattens out.
The lower bound $|\vec{A}+\vec{B}| \ge \big||\vec{A}|-|\vec{B}|\big|$ is reached only when $\vec{A}$ and $\vec{B}$ point in exactly opposite directions (the angle is $180^\circ$ ).
For any angle in between, the resultant magnitude lies strictly between these two limits.

This is the vector form of the ordinary triangle inequality $|x+y| \le |x|+|y|$ you may already know for numbers: here $x$ and $y$ become vectors, and "the sides of a triangle" become "a vector, another vector, and their sum."

Why It Matters in Kinematics

This bound is genuinely useful when combining physical quantities in the plane:

Relative velocity: if a boat has speed 5 m/s relative to water and the river flows at 3 m/s, the boat's speed relative to the ground must lie between $|5-3|=2$ m/s and $5+3=8$ m/s, depending on the angle the boat is steered — it can never be less than 2 or more than 8.
Combining forces or displacements: if you know only the magnitudes of two vectors, not the angle between them, this inequality instantly tells you the range of possible resultant magnitudes without doing any trigonometry. …

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Use the triangle law: if two vectors are drawn head-to-tail, the vector from the first tail to the last head is their sum; if the three arrows form a closed loop, they sum to zero. Applying this to each orientation gives (a) $\to$ (iv), (b) $\to$ (iii), (c) $\to$ (i), (d) $\to$ (ii).

Concept

Triangle law of vector addition: place $\vec{P}$ and $\vec{Q}$ head-to-tail; the single arrow drawn from the tail of $\vec{P}$ to the head of $\vec{Q}$ equals $\vec{P}+\vec{Q}$ . If instead the three vectors are drawn head-to-tail and close back on the starting point, their sum is the zero vector.

Matching each diagram

(i): $\vec{a}$ (up) and $\vec{c}$ (right) are head-to-tail, and $\vec{b}$ goes from the common tail to the final head, so $\vec{b} = \vec{a} + \vec{c}$ . Rearranging, $\vec{b} - \vec{a} = \vec{c}$ — this is relation (c).
(ii): $\vec{a}$ (right), $\vec{b}$ (up) and $\vec{c}$ (back to the origin) form a closed loop with no leftover resultant, so $\vec{a} + \vec{b} + \vec{c} = 0$ — this is relation (d). …

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Q.Column I gives four relations between the vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ ; Column II gives four orientations of these vectors drawn as triangles in the $XY$ -plane. Match each relation to its correct orientation. Column I:

Concept understanding — Triangle Inequality

Triangle Inequality (Bounding the Resultant of Two Vectors)

The Intuition

The Precise Statement

Why It Matters in Kinematics

Concept

Matching each diagram

Concept: Assign Explicit Cartesian Components Instead of Reasoning Purely From the Triangle Law

Steps

Q.Column I gives four relations between the vectors a⃗\vec{a}a, b⃗\vec{b}b and c⃗\vec{c}c; Column II gives four orientations of these vectors drawn as triangles in the XYXYXY-plane. Match each relation to its correct orientation. Column I:

Concept

Matching each diagram

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Q.Column I gives four relations between the vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ ; Column II gives four orientations of these vectors drawn as triangles in the $XY$ -plane. Match each relation to its correct orientation. Column I:

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