MHT-CET Physics · Motion in a Plane

Vectors: Addition and Products

Vectors add by components or by the parallelogram law, R² = A² + B² + 2AB cos θ; the dot product A·B = AB cos θ is zero for perpendicular vectors, and the cross product has magnitude AB sin θ and points perpendicular to both.

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Why this matters

10 PYQs, 2 of them HARD. Five add vectors — a unit vector along a sum, three vectors that may or may not close a triangle, a sum perpendicular to a difference, a resultant that changes when one vector doubles. Five use products: the dot product from the cross product, perpendicularity, and the angle between a sum and a cross product. Two cards.

Concept 1 of 2: Adding Vectors

Add component by component; the unit vector along the result is the result divided by its magnitude. Geometrically, R² = A² + B² + 2AB cos θ. Three vectors form a triangle only if they close — one equals the sum of the other two — which also means they lie in a plane. If the sum of two vectors is perpendicular to their difference, (A + B)·(A − B) = A² − B² = 0, so they have equal magnitudes. A condition such as 'A + 2B is perpendicular to A' gives A + 2B cos θ = 0, which substituted into R² simplifies directly.

Definition

Components: $\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$ ; $\hat{R} = \dfrac{\vec{R}}{|\vec{R}|}$ .
$R^2 = A^2 + B^2 + 2AB\cos\theta$ .
Triangle: one vector is the sum of the other two (non-zero triple product ⇒ no triangle).
$(\vec{A} + \vec{B}) \perp (\vec{A} - \vec{B})$ ⇒ $|\vec{A}| = |\vec{B}|$ .
$\vec{A} + 2\vec{B} \perp \vec{A}$ ⇒ $A = -2B\cos\theta$ ⇒ original resultant $C = B$ .

Resultant

R^2 = A^2 + B^2 + 2AB\cos\theta

Worked example

Unit vector along the sum of

2\hat{i} + \hat{j}

and

\hat{i} - \hat{j} + 2\hat{k}

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 19 April Shift II · Q1Hard

Example 1 · Motion in a Plane · Vector Operations and Components

The resultant of two vectors

\overrightarrow{A}

and

\overrightarrow{B}

\overrightarrow{C}

. If the magnitude of

\overrightarrow{B}

is doubled, the new resultant vector becomes perpendicular to

\overrightarrow{A}

, then the magnitude of

\overrightarrow{C}

Any three vectors form a triangle

Only if they close. Check whether one is the sum of the other two; if not, they form no triangle whatever their lengths.

Drill 4 more on adding vectors

Concept 2 of 2: Dot and Cross Products

A·B = AB cos θ = AₓBₓ + A_yB_y + A_zB_z: zero exactly when the vectors are perpendicular, which turns 'perpendicular' into an equation. |A × B| = AB sin θ, and A × B is perpendicular to both A and B. Given the magnitudes and |A × B|, find sin θ, then cos θ, then A·B. A vector lying in the plane of A and B is perpendicular to A × B.

Definition

$\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$ ; perpendicular ⇒ 0.
$|\vec{A}\times\vec{B}| = AB\sin\theta$ ( $5\sqrt{3}$ , 10 at 30° ⇒ $25\sqrt{3}$ ).
$\vec{A}\times\vec{B}$ is perpendicular to $\vec{A}$ , to $\vec{B}$ , and to any combination of them.
$|a| = \sqrt{26}$ , $|b| = 7$ , $|a\times b| = 35$ ⇒ $a\cdot b = 7$ .

Products

\vec{A}\cdot\vec{B} = AB\cos\theta, \qquad |\vec{A}\times\vec{B}| = AB\sin\theta

Worked example

For which

m

are

2\hat{i} + m\hat{j} - \hat{k}

and

\hat{i} - 3\hat{j} + \hat{k}

perpendicular?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 26 April Shift I · Q11Moderate

Example 2 · Motion in a Plane · Vector Operations and Components

|\overrightarrow{a}| =\sqrt{26},|\overrightarrow{b}| = 7

|\overrightarrow{a}\times\overrightarrow{b}| = 35

, find

\overrightarrow{a}\cdot\overrightarrow{b}

Using sin for the dot product

Dot uses cos θ, cross uses sin θ. At 30°, AB sin θ is half of AB; AB cos θ is (√3/2)AB.

Expecting a cross product to lie in the plane of its vectors

A × B is perpendicular to both A and B, so any sum of A, B and their multiples is perpendicular to it — the angle is 90°, whatever the numbers.

Drill 4 more on dot and cross products

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Adding Vectors
Resultant
$R^2 = A^2 + B^2 + 2AB\cos\theta$
Dot and Cross Products
Products
$\vec{A}\cdot\vec{B} = AB\cos\theta, \qquad |\vec{A}\times\vec{B}| = AB\sin\theta$

Watch out for (3)

Any three vectors form a triangle
→ Adding Vectors
Using sin for the dot product
→ Dot and Cross Products
Expecting a cross product to lie in the plane of its vectors
→ Dot and Cross Products

Test yourself on Motion in a Plane

15 past MHT-CET questions from this chapter, timed at 14 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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