Aspire Faculty ID #11957 · Topic: NIMCET 2025 · Just now
NIMCET 2025

If $\vec{a}, \vec{b}$ and $\vec{c} $ are three vectors such that $\vec{a} \times \vec{b}=\vec{c}$ , $\vec{a}.\vec{c} = 2$ and $\vec{b}.\vec{c} = 1$. If $|\vec{b}| = 1$, then the value of $|\vec{a}| $ is

Solution

Given conditions: $\vec{a} \times \vec{b} = \vec{c}$ 
$\vec{a} \cdot \vec{c} = 2$ 
$\vec{b} \cdot \vec{c} = 1$ 
$|\vec{b}| = 1$ 

Since $\vec{c} = \vec{a} \times \vec{b}$, 
we have: $|\vec{c}| = |\vec{a}||\vec{b}| \sin\theta = |\vec{a}|\sin\theta$ (because $|\vec{b}| = 1$) 

Also: $\vec{a} \cdot \vec{c} = \vec{a} \cdot (\vec{a} \times \vec{b}) = 0$ 
But given: $\vec{a} \cdot \vec{c} = 2$ 

This is possible only if $\vec{c}$ is not perpendicular to $\vec{a}$, meaning $\vec{c}$ is not just $\vec{a} \times \vec{b}$ but also has a component along $\vec{a}$. 
Use the identity: $(\vec{a} \times \vec{b}) \cdot \vec{c} = \det(\vec{a},\vec{b},\vec{c})$ 
But we need magnitudes. Take dot product of $\vec{b}$ with $\vec{c}$: 
$\vec{b} \cdot \vec{c} = \vec{b} \cdot (\vec{a} \times \vec{b}) = 0$ 
But given $\vec{b} \cdot \vec{c} = 1$ 

So again $\vec{c}$ has components outside the perpendicular direction $\vec{c}$ is independent. 

Use the vector triple product identity: $\vec{c} = \vec{a} \times \vec{b}$ 

Take magnitude squared: 
$|\vec{c}|^2 = |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2$ 

Since $|\vec{b}| = 1$: $|\vec{c}|^2 = |\vec{a}|^2 - (\vec{a} \cdot \vec{b})^2$ 

Now use the given dot products. 
We know: $\vec{a} \cdot \vec{c} = 2$ $\vec{b} \cdot \vec{c} = 1$ 

Take magnitude squared of $\vec{c}$: $|\vec{c}|^2 = (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{b})$ 

But also: $|\vec{c}|^2 = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{c}) = 2 \cdot 1 = 2$ 

So: $|\vec{c}|^2 = 2$ 
Thus: $2 = |\vec{a}|^2 - (\vec{a} \cdot \vec{b})^2$ 
Also: $\vec{a} \cdot \vec{b} = \dfrac{\vec{a} \cdot \vec{c}}{|\vec{c}|} = \dfrac{2}{\sqrt{2}} = \sqrt{2}$ 

Therefore: $(\vec{a} \cdot \vec{b})^2 = 2$ 

Substitute: $2 = |\vec{a}|^2 - 2$ 
So: $|\vec{a}|^2 = 4$ 
Hence: $|\vec{a}| = 2$
← Previous Next →

Previous 10 Questions — NIMCET 2025

Nearest first
1
Let A = {1,2,3, ... , 20}. Let $R\subseteq A\times A$ such that R = {(x,y): y = 2x - 7}. Then the number of elements in…
Topic: NIMCET 2025
 2
The number of all even integers between 99 and 999 which are not multiple of 3 and 5 is
Topic: NIMCET 2025
 3
There are two coins, say blue and red. For blue coin, probability of getting head is 0.99 and for red coin, it is 0.01.…
Topic: NIMCET 2025
 4
The value of $\frac{d}{dx}\int ^{2\sin x}_{\sin {x}^2}{e}^{{t}^2}dt$ at $x=\pi$
Topic: NIMCET 2025
 5
The captains of five cricket teams, including India and Australia, are lined up randomly next to one other for a group …
Topic: NIMCET 2025
 6
Let A and B be two square matrices of same order satisfying $A^2+5A+5I =0$ and $B^2+3B+I=0$ repectively. Where I is the…
Topic: NIMCET 2025
 7
Number of permutations of the letters of the word BANGLORE such that the string ANGLE appears together in all permutati…
Topic: NIMCET 2025
 8
The scores of students in a national level examination are normally distributed with a mean of 500 and a standard devia…
Topic: NIMCET 2025
 9
The curve $y=\frac{x}{1+x\tan x}$ attains maxima
Topic: NIMCET 2025
 10
Consider the matrix $$B=\begin{pmatrix}{-1} & {-1} & {2} \\ {0} & {-1} & {-1} \\ {0} & {0} & {-…
Topic: NIMCET 2025

Next 10 Questions — NIMCET 2025

Ascending by ID
1
If x, y and z are three cube roots of 27, then  the determinant of the matrix $\begin{bmatrix}{x} & {y} &…
Topic: NIMCET 2025
 2
Let $A=\{{5}^n-4n-1\colon n\in N\}$ and $B=\{{}16(n-1)\colon n\in N\}$ be sets. Then
Topic: NIMCET 2025
 3
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be unit vectors such that the angle between them is ${\cos }^{-1}\Bigg…
Topic: NIMCET 2025
 4
A tower subtends angles $\alpha, 2\alpha$ and $3\alpha$ respectively at points A, B and C which are lying on a horizont…
Topic: NIMCET 2025
 5
If $\vec{a}$ and $\vec{b}$ are twp vectors such that |$\vec{a}$|=3, |$\vec{b}$|=4 and |$\vec{a}+\vec{b}$|=1, then the v…
Topic: NIMCET 2025
 6
If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}=\hat{i}-2\hat{j}+\hat{k}$, then …
Topic: NIMCET 2025
 7
Consider the sample space $\Omega={\{(x,y):x,y\in{\{1,2,3,4\}\}}}$ where each outcome is equally likely. Let A = {x ≥ 2…
Topic: NIMCET 2025
 8
Let the line $\frac{x}{4}+\frac{y}{2}=1$ meets the x-axis and y-axis at A and B, respectively. M is the midpoint of sid…
Topic: NIMCET 2025
 9
Which one of the following is NOT a correct statement?
Topic: NIMCET 2025
 10
An equilateral triangle is inscribed in the parabola $y^2 = x$. One vertex of the triangle is at the vertex of the para…
Topic: NIMCET 2025
Ask Your Question or Put Your Review.

loading...