AS/A Level Physics Scalars and vectors Study Notes 2025-2027 Syllabus

Addition and Subtraction of Coplanar Vectors

Coplanar vectors are vectors that lie in the same plane. Adding or subtracting coplanar vectors allows us to determine the resultant (single equivalent) vector that represents their combined effect.

Key Idea:

• Vector addition combines two or more vectors to find their resultant.
• Vector subtraction finds the difference between two vectors — equivalent to adding one vector to the negative of another.
• Coplanar vector addition follows geometric laws such as the triangle law and parallelogram law.

1.  Triangle Law of Vector Addition

Statement: If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their resultant is represented in magnitude and direction by the third side of the triangle taken in the opposite order.

Mathematically:

\( \mathrm{\vec{R} = \vec{A} + \vec{B}} \)

Graphical Representation:

• Draw \( \mathrm{\vec{A}} \) to scale.
• From the head of \( \mathrm{\vec{A}} \), draw \( \mathrm{\vec{B}} \) to scale in its direction.
• The resultant \( \mathrm{\vec{R}} \) is drawn from the tail of \( \mathrm{\vec{A}} \) to the head of \( \mathrm{\vec{B}} \).

2.  Parallelogram Law of Vector Addition

Statement: If two vectors are represented by the adjacent sides of a parallelogram, their resultant is represented by the diagonal of the parallelogram drawn from the same point.

Formula for Magnitude:

\( \mathrm{R = \sqrt{A^2 + B^2 + 2AB\cos\theta}} \)

where:

• \( \mathrm{R} \) = magnitude of resultant vector
• \( \mathrm{A, B} \) = magnitudes of the two vectors
• \( \mathrm{\theta} \) = angle between \( \mathrm{\vec{A}} \) and \( \mathrm{\vec{B}} \)

Direction of Resultant:

\( \mathrm{\tan\beta  = \dfrac{B\sin\theta}{A + B\cos\theta}} \)

where \( \mathrm{\beta } \) is the angle between \( \mathrm{\vec{A}} \) and \( \mathrm{\vec{R}} \).

3. Subtraction of Vectors

Rule: To subtract one vector from another, reverse the direction of the vector to be subtracted and then add it using the triangle law.

\( \mathrm{\vec{R} = \vec{A} – \vec{B} = \vec{A} + (-\vec{B})} \)

• Here \( \mathrm{-\vec{B}} \) has the same magnitude as \( \mathrm{\vec{B}} \) but points in the opposite direction.
• Graphically, draw \( \mathrm{\vec{A}} \), then from its head draw \( \mathrm{-\vec{B}} \).
• The resultant \( \mathrm{\vec{R}} \) is drawn from the tail of \( \mathrm{\vec{A}} \) to the head of \( \mathrm{-\vec{B}} \).

Special Cases:

• If \( \mathrm{\theta = 0^\circ} \): vectors act in the same direction → \( \mathrm{R = A + B} \).
• If \( \mathrm{\theta = 180^\circ} \): vectors act in opposite directions → \( \mathrm{R = |A – B|} \).
• If \( \mathrm{\theta = 90^\circ} \): vectors are perpendicular → \( \mathrm{R = \sqrt{A^2 + B^2}} \).