Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A}=2 \hat{i}+3 n \hat{j}+2 \hat{k}$ and $\bar{B}=2 \hat{i}-2 \hat{j}+4 p \hat{k}$, respectively. If both the vectors are at right angle to each other, the value of $n^{-1}$ is ________ .
Explanation:
We are given two conditions:
The particles are at the same distance from the origin. This means that the magnitudes of their position vectors are equal.
The vectors are perpendicular (at right angles) to each other. This means their dot product is zero.
Let’s work through these step-by-step.
● The vectors are given by:
$\vec{A} = 2\hat{i} + 3n \hat{j} + 2\hat{k},$
$\vec{B} = 2\hat{i} -2\hat{j} + 4p \hat{k}.$
● Since the vectors are perpendicular, their dot product must be zero:
$\vec{A} \cdot \vec{B} = (2)(2) + (3n)(-2) + (2)(4p) = 4 - 6n + 8p = 0.$
This can be rearranged to:
$8p = 6n - 4 \quad \Longrightarrow \quad p = \frac{3n - 2}{4} \quad \text{(Equation 1)}.$
● Since the distances from the origin are equal, the magnitudes of the vectors must be equal. Compute the squares of the magnitudes:
For $\vec{A}: \quad |\vec{A}|^2 = 2^2 + (3n)^2 + 2^2 = 4 + 9n^2 + 4 = 8 + 9n^2.$
For $\vec{B}: \quad |\vec{B}|^2 = 2^2 + (-2)^2 + (4p)^2 = 4 + 4 + 16p^2 = 8 + 16p^2.$
Setting them equal:
$8 + 9n^2 = 8 + 16p^2 \quad \Longrightarrow \quad 9n^2 = 16p^2.$
This simplifies to:
$p^2 = \frac{9n^2}{16} \quad \Longrightarrow \quad p = \pm \frac{3n}{4} \quad \text{(Equation 2)}.$
● Now equate the two expressions for $p$ from Equation 1 and Equation 2.
Case 1: Assume $p = \frac{3n}{4}.$
Then,
$\frac{3n - 2}{4} = \frac{3n}{4} \quad \Longrightarrow \quad 3n - 2 = 3n \quad \Longrightarrow \quad -2 = 0,$
which is a contradiction.
Case 2: Assume $p = -\frac{3n}{4}.$
Then,
$\frac{3n - 2}{4} = -\frac{3n}{4} \quad \Longrightarrow \quad 3n - 2 = -3n.$
Simplify by adding $3n$ to both sides:
$6n - 2 = 0 \quad \Longrightarrow \quad 6n = 2 \quad \Longrightarrow \quad n = \frac{1}{3}.$
● Since we found $n = \frac{1}{3},$ its reciprocal is:
$n^{-1} = 3.$
Thus, the value of $n^{-1}$ is $3.$
If the component of the vector $\mathbf{A}$ along the vector $\mathbf{B}$ is twice the component of $\mathbf{B}$ along $\mathbf{A}$, then the ratio of magnitudes of vectors $\mathbf{A}$ and $\mathbf{B}$ is
$1: 2$
$3: 2$
$2: 1$
$3: 1$
Three vectors each of magnitude $3 \sqrt{1.5}$ units are acting at a point. If the angle between any two vectors is $\frac{\pi}{3}$, then the magnitude of the resultant vector of the three vector is
$9 \sqrt{3}$ units
9 units
$\sqrt{6}$ units
3 units
A vector perpendicular to the vector $(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}})$ is
$4 \hat{i}+3 \hat{j}$
$6 \hat{i}$
$3 \hat{i}-4 \hat{j}$
$7 \hat{\mathbf{k}}$
If $\alpha, \beta$ and $\gamma$ are the angles made by a vector with $x, y$ and $z$ axes respectively, then $\sin ^2 \alpha+\sin ^2 \beta=$
$\sin ^2 \gamma$
$\cos ^2 \gamma$
$1+\cos ^2 \gamma$
$1+\sin ^2 \gamma$
If the magnitude of a vector $\mathbf{P}$ is 25 units and its $y$-component is 7 units, then its $x$-component is
24 units
18 units
32 units
16 units
The magnitudes of two vectors are $A$ and $B(A>B)$. If the maximum resultant magnitude of the two vectors is ' $n$ ' times their minimum resultant magnitude, then $\frac{A}{B}=$
$\frac{n}{n-1}$
$\frac{n+1}{n}$
$\frac{n^2+1}{n-1}$
$\frac{n+1}{n-1}$
The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to $\vec{A}$ and its magnitude is half that of $\vec{B}$. The angle between vectors $\vec{A}$ and $\vec{B}$ is _________$^\circ$.
Explanation:
To solve this problem, we'll analyze the conditions given about the vectors $\vec{A}$ and $\vec{B}$, and their resultant $\vec{R}$.
Given:
- $\vec{R} = \vec{A} + \vec{B}$ is perpendicular to $\vec{A}$.
- The magnitude of $\vec{R}$ is half that of $\vec{B}$: $|\vec{R}| = \frac{1}{2}|\vec{B}|$.
Approach:
We know that if $\vec{R}$ is perpendicular to $\vec{A}$, then their dot product is zero:
$ \vec{R} \cdot \vec{A} = 0 $
Substitute $\vec{R} = \vec{A} + \vec{B}$:
$ (\vec{A} + \vec{B}) \cdot \vec{A} = 0 $
$ \Rightarrow $$ \vec{A} \cdot \vec{A} + \vec{B} \cdot \vec{A} = 0 $
$ \Rightarrow $$ |\vec{A}|^2 + |\vec{A}||\vec{B}|\cos \theta = 0 $
$ \Rightarrow $$ \cos \theta = -\frac{|\vec{A}|^2}{|\vec{A}||\vec{B}|} $
$ \Rightarrow $$ \cos \theta = -\frac{|\vec{A}|}{|\vec{B}|} $
Next, we use the second condition involving magnitudes:
$ |\vec{R}| = |\vec{A} + \vec{B}| = \frac{1}{2}|\vec{B}| $
$ \Rightarrow $$ \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos \theta} = \frac{1}{2}|\vec{B}| $
$ \Rightarrow $$ |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\frac{|\vec{A}|}{|\vec{B}|} = \frac{1}{4}|\vec{B}|^2 $
$ \Rightarrow $$ |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}|^2 = \frac{1}{4}|\vec{B}|^2 $
$ \Rightarrow $$ |\vec{B}|^2 - |\vec{A}|^2 = \frac{1}{4}|\vec{B}|^2 $
$ \Rightarrow $$ \frac{3}{4}|\vec{B}|^2 = |\vec{A}|^2 $
$ \Rightarrow $$ |\vec{A}| = \frac{\sqrt{3}}{2}|\vec{B}| $
Finally, substitute this back into the cosine formula:
$ \cos \theta = -\frac{\frac{\sqrt{3}}{2}|\vec{B}|}{|\vec{B}|} $
$ \cos \theta = -\frac{\sqrt{3}}{2} $
This value of $\cos \theta$ corresponds to an angle of $150^\circ$ because $\cos 150^\circ = -\frac{\sqrt{3}}{2}$.
Answer:
The angle between $\vec{A}$ and $\vec{B}$ is $150^\circ$.
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 $\mathrm{n}$ is _________.
Explanation:
To solve this problem, we will use the concepts of vector addition and magnitudes involving the dot product. Given the angle between $\vec{a}$ and $\vec{b}$ is $\cos^{-1}\left(\frac{5}{9}\right)$, we can use the properties of dot products and magnitudes to find the required integer value of $n$.
First, let's use the condition $|\vec{a} + \vec{b}| = \sqrt{2} |\vec{a} - \vec{b}|$. Square both sides to remove the square roots:
$ |\vec{a} + \vec{b}|^2 = 2 |\vec{a} - \vec{b}|^2 $
Now we will expand both sides using the formula for the magnitude of the sum and difference of vectors:
$ |\vec{a} + \vec{b}|^2 = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} $
And
$ |\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} $
Substitute these back into the original equation:
$ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 2 (\vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b}) $
Expand the right-hand side:
$ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 2 \vec{a} \cdot \vec{a} - 4 \vec{a} \cdot \vec{b} + 2 \vec{b} \cdot \vec{b} $
Rearrange all terms to one side to combine like terms:
$ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} - 2 \vec{a} \cdot \vec{a} + 4 \vec{a} \cdot \vec{b} - 2 \vec{b} \cdot \vec{b} = 0 $
Combine like terms:
$ - \vec{a} \cdot \vec{a} + 6 \vec{a} \cdot \vec{b} - \vec{b} \cdot \vec{b} = 0 $
We know that:
$ \vec{a} \cdot \vec{a} = |\vec{a}|^2 \quad \text{and} \quad \vec{b} \cdot \vec{b} = |\vec{b}|^2 \quad \text{and} \quad \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) $
Since the angle between $\vec{a}$ and $\vec{b}$ is $\cos^{-1}\left(\frac{5}{9}\right)$, we have:
$ \cos(\theta) = \frac{5}{9} $
Substitute these back into the equation:
$ -|\vec{a}|^2 + 6 |\vec{a}| |\vec{b}| \left(\frac{5}{9}\right) - |\vec{b}|^2 = 0 $
Simplify it further:
$ -|\vec{a}|^2 + \frac{30}{9} |\vec{a}| |\vec{b}| - |\vec{b}|^2 = 0 $
$ -|\vec{a}|^2 + \frac{10}{3} |\vec{a}| |\vec{b}| - |\vec{b}|^2 = 0 $
We know that $|\vec{a}| = n |\vec{b}|$. Substitute this into the equation:
$ -(n |\vec{b}|)^2 + \frac{10}{3} (n |\vec{b}|) |\vec{b}| - |\vec{b}|^2 = 0 $
Simplify it:
$ -n^2 |\vec{b}|^2 + \frac{10}{3} n |\vec{b}|^2 - |\vec{b}|^2 = 0 $
Factor out $|\vec{b}|^2$:
$ |\vec{b}|^2 \left(-n^2 + \frac{10}{3} n - 1\right) = 0 $
Since $|\vec{b}|^2 \neq 0$, we can solve:
$ -n^2 + \frac{10}{3} n - 1 = 0 $
Multiply through by 3 to clear the fraction:
$ -3n^2 + 10n - 3 = 0 $
Rearrange it to match standard quadratic form:
$ 3n^2 - 10n + 3 = 0 $
Solve this quadratic equation using the quadratic formula:
$ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $
Here, $a = 3$, $b = -10$, and $c = 3$:
$ n = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} = \frac{10 \pm \sqrt{100 - 36}}{6} = \frac{10 \pm \sqrt{64}}{6} = \frac{10 \pm 8}{6} $
This results in two possible solutions for $n$:
$ n = \frac{18}{6} = 3 \quad \text{and} \quad n = \frac{2}{6} = \frac{1}{3} $
However, since $n$ is given to be an integer, we take:
$ \boxed{3} $
Three vectors $\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{OR}}$ each of magnitude $\mathrm{A}$ are acting as shown in figure. The resultant of the three vectors is $\mathrm{A} \sqrt{x}$. The value of $x$ is _________.
Explanation:
$\begin{aligned} &\begin{aligned} & (\vec{P}+\vec{Q})=\sqrt{2} A \\ & R=A \\ & \theta=90^{\circ} \end{aligned}\\ &\text { Resultant }=\sqrt{3} A \end{aligned}$
For three vectors $\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k}), \vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k})$ and $\vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k})$, if $\vec{A} \cdot(\vec{B} \times \vec{C})=0$, then value of $x$ is ________.
Explanation:
To determine the value of $ x $, given the vectors $\vec{A}=(-x \hat{i} - 6 \hat{j} - 2 \hat{k})$, $\vec{B}=(-\hat{i} + 4 \hat{j} + 3 \hat{k})$, and $\vec{C}=(-8 \hat{i} - \hat{j} + 3 \hat{k})$, and the condition $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, we proceed as follows:
First, we calculate the cross product $\vec{B} \times \vec{C}$:
$ \vec{B} \times \vec{C} = 15 \hat{i} - 21 \hat{j} + 33 \hat{k} $
Next, using the condition $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, we compute the dot product:
$ \vec{A} \cdot (\vec{B} \times \vec{C}) = (-x \hat{i} - 6 \hat{j} - 2 \hat{k}) \cdot (15 \hat{i} - 21 \hat{j} + 33 \hat{k}) $
$ \Rightarrow (-x)(15) + (-6)(-21) + (-2)(33) = 0 $
$ \Rightarrow -15x + 126 - 66 = 0 $
Solving this equation for $ x $:
$ -15x + 60 = 0 $
$ \Rightarrow x = 4 $
Thus, the value of $ x $ is $ 4 $.
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 $x$ and 3 respectively where $x=$ _________.
Explanation:
To find a vector that has the same magnitude as vector $\vec{A}$ and is parallel to vector $\vec{B}$, we use the formula:
$\vec{N} = |\vec{A}| \hat{B}$
First, let's find the magnitude of $\vec{A}$, which is $|\vec{A}|$.
$|\vec{A}| = \sqrt{3^2 + 4^2}=5$
We're given that $\vec{B} = 4 \hat{i}+3 \hat{j}$, so to make $\vec{N}$ parallel to $\vec{B}$ and have it have the same magnitude as $\vec{A}$, they should be the same when $\vec{N}$'s magnitude is divided by 5, since $|\vec{A}|=5$.
So, $\vec{N} = \frac{5(4\hat{i}+3\hat{j})}{5} = 4\hat{i}+3\hat{j}$.
This means the $x$ component of the vector in the first quadrant is 4, and the $y$ component is given as 3. So, $x=4$.
The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q}+2 \vec{P})$ and $(2 \vec{Q}-2 \vec{P})$ is :
If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at angle $\theta$, then
If $\overrightarrow P = 3\widehat i + \sqrt 3 \widehat j + 2\widehat k$ and $\overrightarrow Q = 4\widehat i + \sqrt 3 \widehat j + 2.5\widehat k$ then, the unit vector in the direction of $\overrightarrow P \times \overrightarrow Q $ is ${1 \over x}\left( {\sqrt 3 \widehat i + \widehat j - 2\sqrt 3 \widehat k} \right)$. The value of $x$ is _________.
Explanation:
$\vec{Q}=4 \hat{i}+\sqrt{3} \hat{j}+2.5 \hat{k}$
$\vec{P} \times \vec{Q}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5\end{array}\right|$
$=\hat{i}\left(\frac{\sqrt{3}}{2}\right)-\hat{j}\left(-\frac{1}{2}\right)+\hat{k}(-\sqrt{3})$
$=\frac{\sqrt{3}}{2} \hat{i}+\frac{\hat{j}}{2}-\sqrt{3} \hat{k}$
$|\vec{P} \times \vec{Q}|=\sqrt{\frac{3}{4}+\frac{1}{4}+3}=2$
Unit vector along $\vec{P} \times \vec{Q}=\frac{1}{4}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k})$
$x=4$
Vectors $a\widehat i + b\widehat j + \widehat k$ and $2\widehat i - 3\widehat j + 4\widehat k$ are perpendicular to each other when $3a + 2b = 7$, the ratio of $a$ to $b$ is ${x \over 2}$. The value of $x$ is ____________.
Explanation:
$ \begin{aligned} & (a \hat{i}+b \hat{j}+\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+4 \hat{k})=0 \\\\ & 2 a-3 b+4=0 \end{aligned} $
On solving, $2 a-3 b=-4$
Also given
$ 3 a+2 b=7 $
We get $\mathrm{a}=1, \mathrm{~b}=2$
$ \begin{aligned} & \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{x}}{2} \Rightarrow \mathrm{x}=\frac{2 \mathrm{a}}{\mathrm{b}}=\frac{2 \times 1}{2} \\\\ & \Rightarrow \mathrm{x} =1 \end{aligned} $
When vector $\vec{A}=2 \hat{i}+3 \hat{j}+2 \hat{k}$ is subtracted from vector $\overrightarrow{\mathrm{B}}$, it gives a vector equal to $2 \hat{j}$. Then the magnitude of vector $\overrightarrow{\mathrm{B}}$ will be :
Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is:
If two vectors $\overrightarrow P = \widehat i + 2m\widehat j + m\widehat k$ and $\overrightarrow Q = 4\widehat i - 2\widehat j + m\widehat k$ are perpendicular to each other. Then, the value of m will be :
The angle between force $\mathbf{F}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and displacement $\mathbf{d}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is
$\cos ^{-1}(0.16)$
$\cos ^{-1}(0.32)$
$\cos ^{-1}(0.24)$
$\cos ^{-1}(0.64)$
If the projection of $2 \hat{i}+4 \hat{j}-2 \hat{k}$ on $\hat{i}+2 \hat{j}+\alpha \hat{k}$ is zero. Then, the value of $\alpha$ will be ___________.
Explanation:
$\overrightarrow A = 2\widehat i + 4\widehat j - 2\widehat k$
$\overrightarrow B = \widehat i + 2\widehat j + \alpha \widehat k$
$\overrightarrow A \,.\,\overrightarrow B = 0$, as $\overrightarrow A $ should be perpendicular to $\overrightarrow B $
$ \Rightarrow 2 + 8 - 2\alpha = 0$
$\alpha = 5$
If $\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k})\, \mathrm{m}$ and $\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \,\mathrm{m}$. The magnitude of component of vector $\vec{A}$ along vector $\vec{B}$ will be ____________ $\mathrm{m}$.
Explanation:
And $ |\vec{B}|=\sqrt{1^2+2^2+2^2}=\sqrt{9}=3 $
Magnitude of component of
$\vec{A}$ along $\vec{B}=\frac{\vec{A} \cdot \vec{B}}{|B|}=\frac{6}{3}=2$
Two vectors $\overrightarrow A $ and $\overrightarrow B $ have equal magnitudes. If magnitude of $\overrightarrow A $ + $\overrightarrow B $ is equal to two times the magnitude of $\overrightarrow A $ $-$ $\overrightarrow B $, then the angle between $\overrightarrow A $ and $\overrightarrow B $ will be :
$\overrightarrow A $ is a vector quantity such that $|\overrightarrow A |$ = non-zero constant. Which of the following expression is true for $\overrightarrow A $ ?
Which of the following relations is true for two unit vector $\widehat A$ and $\widehat B$ making an angle $\theta$ to each other?
An ant starts from the origin and crawls 10 cm along the $X$-axis and then 20 cm along the $Y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $X$-axis and has a magnitude of $\sqrt{2} \mathrm{~cm}$ is
30 cm
$30 \sqrt{2} \mathrm{~cm}$
$\frac{30}{\sqrt{2}} \mathrm{~cm}$
15 cm
The component of a vector $\mathbf{P}=3 \hat{i}+4 \hat{j}$ along the direction $(\hat{i}+2 \hat{j})$ is
If two vectors $\mathbf{A}$ and $\mathbf{B}$ are mutually perpendicular, then the component of $\mathbf{A}-\mathbf{B}$ along the direction of $\mathbf{A}+\mathbf{B}$ is
Which of the following is not true about vectors $\mathbf{A}, \mathbf{B}$ and $\mathbf{C}$ ?
Explanation:
To solve the problem, we need to determine the vectors normal to the planes containing given vectors and find the angle between those normal vectors.
First, let's find the vectors normal to the plane containing vectors $\overrightarrow{A}$ and $\overrightarrow{B}$. The normal vector can be calculated using the cross product:
$\overrightarrow{A} \times \overrightarrow{B} = (\widehat{i} + \widehat{j}) \times (\widehat{j} + \widehat{k})$.
Using the properties of the cross product:
$\begin{aligned} \overrightarrow{A} \times \overrightarrow{B} &= (\widehat{i} \times \widehat{j}) + (\widehat{i} \times \widehat{k}) + (\widehat{j} \times \widehat{j}) + (\widehat{j} \times \widehat{k}) \\ &= \widehat{k} - \widehat{j} + 0 + \widehat{i} \\ &= \widehat{i} - \widehat{j} + \widehat{k}. \end{aligned}$
So, the vector normal to the plane containing vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\overrightarrow{N}_{1} = \widehat{i} - \widehat{j} + \widehat{k}$.
Next, let's find the vector normal to the plane containing vectors $\overrightarrow{A}$ and $\overrightarrow{C}$:
$\overrightarrow{A} \times \overrightarrow{C} = (\widehat{i} + \widehat{j}) \times (-\widehat{i} + \widehat{j})$.
Using the properties of the cross product:
$\begin{aligned} \overrightarrow{A} \times \overrightarrow{C} &= (\widehat{i} \times -\widehat{i}) + (\widehat{i} \times \widehat{j}) + (\widehat{j} \times -\widehat{i}) + (\widehat{j} \times \widehat{j}) \\ &= 0 + \widehat{k} - \widehat{k} + 0 \\ &= \widehat{k} - \widehat{k} + 0 \\ &= 2 \widehat{k}. \end{aligned}$
So, the vector normal to the plane containing vectors $\overrightarrow{A}$ and $\overrightarrow{C}$ is $\overrightarrow{N}_{2} = 2 \widehat{k}$. But we just need the direction of this vector, not its magnitude, so we can simplify it to $\widehat{k}$.
Now, to find the angle between the two normal vectors $\overrightarrow{N}_{1} = \widehat{i} - \widehat{j} + \widehat{k}$ and $\overrightarrow{N}_{2} = \widehat{k}$, we use the dot product formula:
$\cos(\theta) = \frac{\overrightarrow{N}_{1} \cdot \overrightarrow{N}_{2}}{||\overrightarrow{N}_{1}|| ||\overrightarrow{N}_{2}||}$
First, calculate the dot product:
$\overrightarrow{N}_{1} \cdot \overrightarrow{N}_{2} = (\widehat{i} - \widehat{j} + \widehat{k}) \cdot \widehat{k} = 0 + 0 + 1 = 1$
Next, find the magnitude of the vectors:
$||\overrightarrow{N}_{1}|| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3}$
$||\overrightarrow{N}_{2}|| = \sqrt{0^2 + 0^2 + 1^2} = 1$
Now substitute these into the cosine formula:
$\cos(\theta) = \frac{1}{\sqrt{3} \times 1} = \frac{1}{\sqrt{3}} = \sqrt{\frac{1}{3}}$
Therefore, $\cos^{-1}(\sqrt{\frac{1}{3}}) = \cos^{-1}(\frac{1}{\sqrt{3}})$ indicates $x = 3$.
So, the value of $x$ is:
3
Explanation:
It is possible only when $\overrightarrow P \times \overrightarrow Q = \overrightarrow Q \times \overrightarrow P $ = 0
$ \Rightarrow $ We know, $\overrightarrow P \times \overrightarrow Q $ = PQsin $\theta $
Only if $\overrightarrow P = 0$
or $\overrightarrow Q = 0$
or sin $\theta $ = 0 $ \Rightarrow $ $\theta $ = 0 or 180o
The angle b/w $\overrightarrow P $ & $\overrightarrow Q $ is $\theta$(0$^\circ$ < $\theta$ < 360$^\circ$)
So, $\theta$ = 180$^\circ$
Two forces $\left( {\overrightarrow P + \overrightarrow Q } \right)$ and $\left( {\overrightarrow P - \overrightarrow Q } \right)$ where $\overrightarrow P \bot \overrightarrow Q $, when act at an angle $\theta$1 to each other, the magnitude of their resultant is $\sqrt {3({P^2} + {Q^2})} $, when they act at an angle $\theta$2, the magnitude of their resultant becomes $\sqrt {2({P^2} + {Q^2})} $. This is possible only when ${\theta _1} < {\theta _2}$.
Statement II :
In the situation given above.
$\theta$1 = 60$^\circ$ and $\theta$2 = 90$^\circ$
In the light of the above statements, choose the most appropriate answer from the options given below :-
[Take $\sqrt 3 = 1.7$, $\sqrt 2 = 1.4$ Given $\widehat i$ and $\widehat j$ unit vectors along x, y axis]
Reason R : Polygon law of vector addition yields $\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {AD} = 2\overrightarrow {AO} $
In the light of the above statements, choose the most appropriate answer from the options given below :
Choose the correct answer from the options given below :
$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF} + \overrightarrow {AG} + \overrightarrow {AH} $,
if, $\overrightarrow {AO} = 2\widehat i + 3\widehat j - 4\widehat k$
One of the rectangular components of a force of 40 N is 20$\sqrt3$ N. What is the other rectangular component?
Explanation:
$\overrightarrow P + \overrightarrow Q = \overrightarrow R $
$ \Rightarrow $ P2 + Q2 + 2PQcos$\theta $ = R2 = P2
[As $\left| {\overrightarrow R } \right| = \left| {\overrightarrow P } \right|$]
$ \Rightarrow $ 2Pcos$\theta $ + Q = 0
tan $\alpha $ = ${{2P\sin \theta } \over {Q + 2P\cos \theta }}$ = $\infty $
$ \Rightarrow $ $\alpha $ = 90o
If $\mathbf{r}_1=2 \hat{\mathbf{x}}, \mathbf{r}_2=2 \hat{\mathbf{y}}$, where $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ are unit vectors along the $X$-axis and $Y$-axis respectively, then the magnitude of $\mathbf{r}_1+\mathbf{r}_2$ is
$2 \sqrt{2}$
$2 \sqrt{3}$
$3 \sqrt{2}$
$\sqrt{3}$
Let $\mathbf{A}_1+\mathbf{A}_2=5 \mathbf{A}_3, \mathbf{A}_1-\mathbf{A}_2=3 \mathbf{A}_3, \mathbf{A}_3=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}$, then $\frac{\left|\mathbf{A}_1\right|}{\left|\mathbf{A}_2\right|}$ is
4
8
2
6
If $0.5 \hat{\mathbf{i}}+0.8 \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is a unit vector, then $c$ is
$\sqrt{0.89}$
0.2
0.3
$\sqrt{0.11}$