Multiplication Of Vectors - Practice Questions & MCQ

UNIT VECTOR

A vector having magnitude of one unit is called unit vector. It is represented by a cap/hat over the letter. Eg- $\hat{R}$ is called as unit vector of $\vec{R}$. Its direction is along the $\vec{R}$ and magnitude is unit.

Unit vector along $\vec{R}$ -

$
\hat{R}=\frac{\vec{R}}{|\vec{R}|}
$

ORTHOGONAL UNIT VECTORS
It is defined as the unit vectors described under the three-dimensional coordinate system along $x, y$, and $z$ axis. The three unit vectors are denoted by $\mathrm{i}, \mathrm{j}$ and k respectively.

Any vector (Let us say $\vec{R}$ ) can be written as-

$
\vec{R}=x \hat{i}+y \hat{j}+z \hat{k}
$

Where $\mathrm{x}, \mathrm{y}$ and z are components of $\vec{R}$ along $\mathrm{x}, \mathrm{y}$ and z direction respectivly.
Magnitude of $\vec{R}$ -

$
|\vec{R}|=\sqrt{x^2+y^2+z^2}
$
 

Unit vector-

1.  If a vector is multiplied by any scalar

   $
   \begin{gathered}
   \vec{Z}=n \cdot \vec{Y} \\
   (n=1,2,3 . .)
   \end{gathered}
   $

   
   Vector $\times$ Scalar $=$ Vector
   We get again a vector.
   2. If a vector is multiplied by any real number (eg 2 or -2 ) then again, we get a vector quantity.
   E.g.
   - If $\vec{A}$ is multiplied by 2 then the direction of the resultant vector is the same as that of the given vector.

   $
   \text { Vector }=2 \vec{A}
   $

   - If $\vec{A}$ is multiplied by (-2), then the direction of the resultant is opposite to that of the given vector.

   $
   \text { Vector }=-2 \vec{A}
   $

   3. Scalar or Dot or Inner Product
   - Scalar product of two vector $\vec{A} \& \vec{B}$ written as $\vec{A} . \vec{B}$
   - $\vec{A} \cdot \vec{B}$ is a scalar quantity given by the product of the magnitude of $\vec{A} \& \vec{B}$ and the cosine of a smaller angle between them.

   $
   \vec{A} \cdot \vec{B}=A B \cdot \cos \Theta
   $
    

       Figure showing the representation of scalar products of vectors.

Important results-

$
\begin{aligned}
& \hat{i} \cdot \hat{j}=\hat{j} \cdot \hat{k}=\hat{k} \cdot \hat{i}=0 \\
& \hat{i} \cdot \hat{i}=\hat{j} \cdot \hat{j}=\hat{k} \cdot \hat{k}=1 \\
& \vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}
\end{aligned}
$

4. Vector or cross product
- Vector or cross product of two vectors $\vec{A} \& \vec{B}$ written as $\vec{A} \times \vec{B}$
- $A \times B$ is a single vector whose magnitude is equal to the product of the magnitude of $\vec{A} \& \vec{B}$ and the sine of the smaller angle $\theta$ between them.
- $\vec{A} \times \vec{B}=A B \sin \theta$

The figure shows the representation of the cross product of vectors.

Important results-

$
\begin{aligned}
\hat{i} & \times \hat{j}=\hat{k}, \hat{j} \times \hat{k}=\hat{i}, \hat{k} \times \hat{i}=\hat{j} \\
\hat{i} & \times \hat{i}=\hat{j} \times \hat{j}=\hat{k} \times \hat{k}=\overrightarrow{0} \\
\vec{A} \times \vec{B} & =-\vec{B} \times \vec{A}
\end{aligned}
$