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MULTIPLICATION OF VECTORS - (Concept)
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-
-
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}
$
| Exam | Chapter |
| JEE MAIN | Kinematics |
When we multiply a vector with a number -2 , we get
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let The magnitude of a coplanar vector
such that is given by
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\text { Angle between }(\hat{l}+\hat{j}) \text { and }(\hat{l}-\hat{j}) \text { is (in degrees) }
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A particle is thrown with 10m/s at an angle of $60^0$ with horizontal. The time at which its velocity is perpendicular to the initial velocity is ( g = 10 m/s )?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\text { If } \vec{a}, \vec{b} \text { are unit vectors such that }(\vec{a}+\vec{b}) \cdot[(2 \vec{a}+3 \vec{b}) \times(3 \vec{a}-2 \vec{b})]=0 \text {, then angle between } \vec{a} \text { and } \vec{b} \text { is - }$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
the value of
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then the magnitude of
is where V is the linear velocity, w is angular velocity and r is the radius vector and
is given by
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The area of the parallelogram formed from the vectors $\vec{A}=\hat{l}-2 \hat{j}+3 \hat{k}$ and $\vec{B}=3 \hat{l}-2 \hat{j}+\hat{k}$ as adjacent side is $4 \sqrt{n}$ units. find the value of n ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The unit vector of is-
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the angle between
and
is
The value of
will be _______
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$ \vec{A}$ is a vector quantity such that $|\vec{A}|=$ non - zero constant. Which of the following expressions is true for $\vec{A}$?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the projection of is zero. Then, the value of
will be _________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What will be the projection of vector $\vec{A}=\hat{i}+\hat{j}+\hat{k}$ on vector $\vec{B}=\hat{i}+\hat{j}$ ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Three particles are moving along the vectors
respectively. They strike a point and start to move in different directions. Now particle
is moving normal to the plane which contains vector
. Similarly, the particle
is moving normally to the plane which contains the vector
. The angle between the direction of motion of
. Then the value of
is___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A unit of vector in the plane of
and
is such that
where
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\vec{a}^{\prime}=\hat{i}+\hat{j}, \quad \vec{b}^{\prime}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{c}=2 \hat{i}+\hat{j}-\hat{k}$. Then altitude of the parallelepiped formed by the vectors
$\vec{a}, \vec{b}, \vec{c} {\text {having base formed by }} \vec{b}$, and $\vec{c}_{\text {is }}\left(\vec{a}, \vec{b}, \vec{c}\right.$ and $\left.\vec{a}^{\prime}, \vec{b}^{\prime}, \vec{c}^{\prime},\right)$ are reciprocal systems of vectors.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be four non-zero vectors such that
then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are non- coplanar vectors and
holds for some 'X' and 'y' then value of ' ' is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If a vector is perpendicular to the vectors
, then the value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The unit vector along is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
. The value of
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
, then the component of
along
is $x\sqrt{3}$ then the value of x is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\text { The vector } \vec{A} \text { is given as } \vec{A}=3 \hat{i}+3 \hat{j} \text {. Find the angle (in degree) that the vector makes with the positive } \mathrm{y} \text {-axis }
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
and
. The value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
. The value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If , then the angle between A and B is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
When we multiply a vector with the number -2 we get
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Choose the correct statement-
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The wavefront of a light beam is given by the equation $x+2 y+3 z=c$, (where c is arbitrary constant) then the angle made by the direction of light with the $y$-axis is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If ;
. Then angle (in degree) between
and
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The truth table for the circuit given in the fig. is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following statements is not true ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A unit vector is represented as . Hence, the value of ‘b’ is $\sqrt{x}$. Find x.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what value of m is the vector perpendicular to
?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If The value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If ,, then the vector having the same magnitude as B and parallel to A is,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A unit vector is represented as $[(0.8 \hat{i}+b \hat{j}+0.4 \hat{k})]$. Hence the value of ‘b’ must be
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are two unit vectors such that
are perpendicular to each other, then angle between
is _____ degrees
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Vectors and
are perpendicular to each other when
, the ratio of a to b is
The value of x is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If two vectors $\tilde{\mathrm{P}}=\hat{\mathrm{i}}+2 \mathrm{~m} \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}$ and $\tilde{\mathrm{Q}}=4 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\mathrm{m} \hat{\mathrm{k}}$ are perpendicular to each other. Then, the value of m will be:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then, The unit vector in the direction of
The value of x is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then curl will be-
| A. |
|
| B. |
|
| C. |
|
| D. |
|
A vector has magnitude same as that of $\vec{A}=3 \hat{i}+4 \hat{j}$ and is parallel to $\vec{B}=4 \hat{i}+3 \hat{j}$. The $x$ and y components of this vector in first quadrant are $\mathrm{x}$ and 3 respectively where $\mathrm{x}=$______.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The correct statement/s about Hydrogen bonding is/are :
A. Hydrogen bonding exists when H is covalently bonded to the highly electro negative atom.
B. Intermolecular H bonding is present in o-nitro phenol
C. Intramolecular H bonding is present in HF.
D. The magnitude of H bonding depends on the physical state of the compound.
E. H-bonding has powerful effect on the structure and properties of compounds.
Choose the correct answer from the options given below :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If a vector $2 \hat{i}+3 \hat{j}+8 \hat{k}$ is perpendicular to the vectors $-4 \hat{i}+4 \hat{j}+\alpha \hat{k}$. Then the value of $\alpha$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\overline{E+\bar{C} D}=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\hat{n}$ is a unit vector in the direction of the vector $\vec{A}$, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\vec{P} \times \vec{Q}=\vec{R}$ then which of the following statement is not true?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The angle between two vectors:
$\vec{A}=3 \hat{i}+4 \hat{j}+5 \hat{k}$ And $\vec{B}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ will be
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The unit vector along $\hat{i}+\hat{j}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following physical quantities is an axial vector?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For three vectors $\overrightarrow{\mathrm{A}}=(-x \hat{\mathrm{i}}-6 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})$, $\vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k}) \quad$ and $\quad \vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k}), \quad$ if $\overrightarrow{\mathrm{A}} \cdot(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}})=0$, them value of $x$ is_____________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The angle between $A=\hat{i}+\hat{j}$ and $B=\hat{i}-\hat{j}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{B}}=|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}|$. Then the value of $|\vec{A}-\vec{B}|_{\text {will be : }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\vec{a}$ and $\vec{b}$ makes an angle $\cos ^{-1}\left(\frac{5}{9}\right)$ with each other, then $|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$ for $|\vec{a}|=n|\vec{b}|$ The integer value of $n$ is ________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
It is found that $|A+B|=|A|$. This necessarily implies,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If a vector is multiplied by a real positive number, then which of the following statement is correct?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The component of a vector r along X-axis will have maximum value if
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If vector $\vec{A}$ is acting along the y-axis, its y-component is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\overrightarrow{\mathrm{A}}=2 \hat{i}-\hat{j}+3 \hat{k}$, then the magnitude of the vector $\overrightarrow{\mathrm{A}}$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\vec{p}=0.5 \hat{i}+0.8 \hat{j}+c \hat{k}$ If $\vec{p}$ is a unit vector then the value of C is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\widehat{p}=0.5 \widehat{i}+0.8 \widehat{j}-c \widehat{k}, \widehat{p}$ is the unit vector. then the value of ' c ' is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\overrightarrow{\mathrm{A}}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is_______
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Force $\vec{F}$ applied on a body is written as $\vec{F}=(\hat{n} \cdot \vec{F}) \hat{n}+\vec{G}$, where $\hat{n}$ is a unit vector. The vector $\vec{G}$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q}+2 \vec{P})$ and $(2 \vec{Q}-2 \vec{P})$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Two particles are located at equal distance from the origin. The position vectors of those are represented by $\overrightarrow{\mathrm{A}}=2 \hat{\mathrm{i}}+3 n \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\vec{B}=2 \hat{i}-2 \hat{j}+4 p \hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $\mathrm{n}^{-1}$ is _____.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A :
