When vectors interact, we learn how they are related in space. Vector multiplication allows us to extract hidden geometric and physical meaning— such as angles, projection, work done, rotation, and area.
Dot Product (Scalar Product)
The dot product of two vectors tells us how much one vector points in the direction of another. The result is a scalar.
Definition
Here, \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).
Physical meaning
- If vectors point in the same direction, dot product is maximum.
- If vectors are perpendicular, dot product is zero.
- If vectors point in opposite directions, dot product is negative.
Dot product using components (Derivation)
Let:
Using orthogonality of unit vectors:
Applications of dot product
- Finding angle between vectors
- Work done: \( W = \vec{F}\cdot\vec{s} \)
- Checking perpendicularity
Solved Examples: Dot Product
Example 1
Find \( \vec{A}\cdot\vec{B} \) if \[ \vec{A} = 3\hat{i} + 4\hat{j}, \quad \vec{B} = 5\hat{i} - 2\hat{j}. \]
Example 2 (Work Done)
A force \[ \vec{F} = 10\hat{i} + 5\hat{j}\ \text{N} \] moves an object by \[ \vec{s} = 2\hat{i}\ \text{m}. \] Find the work done.
Cross Product (Vector Product)
The cross product of two vectors gives a vector that is perpendicular to the plane containing the two vectors.
Definition
Here, \( \hat{n} \) is a unit vector perpendicular to both \( \vec{A} \) and \( \vec{B} \), given by the right-hand rule.
Physical meaning
- Magnitude gives area of parallelogram
- Direction gives axis of rotation
- Zero if vectors are parallel
Cross product using components (Derivation)
Applications of cross product
- Torque: \( \vec{\tau} = \vec{r}\times\vec{F} \)
- Angular momentum: \( \vec{L} = \vec{r}\times\vec{p} \)
- Area of triangle and parallelogram
Solved Examples: Cross Product
Example 3
Find \( \vec{A}\times\vec{B} \) if \[ \vec{A} = \hat{i} + 2\hat{j}, \quad \vec{B} = 3\hat{i} + \hat{j}. \]
Example 4 (Area)
Find the area of a parallelogram formed by vectors \[ \vec{A} = 4\hat{i}, \quad \vec{B} = 3\hat{j}. \]
Practice Questions
Conceptual
- Why is dot product zero for perpendicular vectors?
- Why is cross product zero for parallel vectors?
- Why does dot product give a scalar but cross product a vector?
Numerical
- Find \( \vec{A}\cdot\vec{B} \) if \( \vec{A} = 2\hat{i} - \hat{j} \), \( \vec{B} = \hat{i} + 3\hat{j} \).
- Find the angle between two vectors using dot product.
- Find torque when \( \vec{r} = 2\hat{i} \) and \( \vec{F} = 3\hat{j} \).
- Find the area of the triangle formed by vectors \( \vec{A} \) and \( \vec{B} \).
Summary
- Dot product → scalar → projection & work
- Cross product → vector → area & rotation
- \( \vec{A}\cdot\vec{B} = |\vec{A}|\,|\vec{B}|\cos\theta \)
- \( |\vec{A}\times\vec{B}| = |\vec{A}|\,|\vec{B}|\sin\theta \)