Scalars and Vectors | Vector Multiplication | Short Questions and Numerical Problem Solutions | Class 11 (Physics) | Physics in Depth

Vector Multiplication

Multiplication of a vector by a scalar

Let $\overrightarrow{a}$ be the vector and $\overrightarrow{k}$ be the scalar, if we multipy them the result is a vector (i.e., k$\overrightarrow{a}$).

Multiplication of a vector by a vector

Vector multiplication are not straightforward as scalar multiplication as they do not follow the ordinary rules of algebra. We can multiply any two vectors in two ways: a. Scalar product and b. Vector product

Scalar product / dot product

Consider any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ inclined at an angle $\theta$ with each other as in fig. Then we define the scalar product of these two vectors as,

$\overrightarrow{a}.\overrightarrow{b}=abcos\theta$ Note that! $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{a}$ Scalar product of any two vectors is always scalar quantity. For example, $P=\overrightarrow{F}.\overrightarrow{v}$. Here, force and velocity are vector quantity but the power which is a scalar product of force and velocity is a scalar quantity.

Geometrical significance / meaning
The scalar product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is the multiplication of magnitude of one vector with the projection of another vector. Here, the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$ is $acos\theta$ and magnitude of $\overrightarrow{b}$ is b and thus the scalar product of this vectors is, $\overrightarrow{a}.\overrightarrow{b}=abcos\theta$
Scalar product of unit vectors
Using the definition $\overrightarrow{a}.\overrightarrow{b}=abcos\theta$, we can find the scalar product of unit vectors as follows: $\hat{i}.\hat{i}=1$ , $\hat{j}.\hat{j}=1$ , $\hat{k}.\hat{k}=1$ (∵ θ =0⁰)
$\hat{i}.\hat{j}=0$, $\hat{j}.\hat{k}=0$, $\hat{k}.\hat{i}=0$ (∵ θ = 90⁰)

Vector product / cross product

Consider any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ inclined at an angle $\theta$ with each other as in fig. Then we define the vector product of these two vectors as,

$\overrightarrow{a}\times \overrightarrow{b}=absin\theta \hat{n}$ where $\hat{n}$ is the unit vector which acts in the direction of $\overrightarrow{a}\times \overrightarrow{b}$ ( i.e., in the direction perpendicular to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$). Note that! $\overrightarrow{a}\times \overrightarrow{b}\ne \overrightarrow{b}\times \overrightarrow{a}$ Vector product of any two vectors is always a vector quantity. For example, $\overrightarrow{\tau }=\overrightarrow{r}.\overrightarrow{F}$. Here, $\overrightarrow{r}$ is the position vector relative to the fixed axis of rotation (Click on Rotational Dynamics for detail) and force are vector quantity and the torque which is a vector product or cross product of position vector and force is a vector quantity.

Vector product of unit vectors

Using the definition, $\overrightarrow{a}.\overrightarrow{b}=absin\theta \hat{n}$, we find the cross product of unit vectors. Vector product of unit vectors follows the cyclic order as in the fig. below. Remember the rule! When you cross product, the two unit vectors along the direction of arrow, the result will be the positive of the new vector along the circle and when you product them in reverse order, the result will be the negative of the new vector.

Geometrical significance / meaning
The magnitude of vector product geometrically signify the area of parallelogram as in figure. The vector product of two vectors i.e., $\overrightarrow{a}\times \overrightarrow{b}$ is normal(perpendicular) to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$.
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