Elementary Operation Transformation Of A Matrix

Step 1: Write the given equation:
The total probability is given as: $P(0)%20%2B%20P(1)%20%2B%20P(2)%20%2B%20P(3)%20%2B%20P(4)%20%3D%201$ Substituting the known values: $0.1%20%2B%20k%20%2B%202k%20%2B%20k%20%2B%200.1%20%3D%201$ where $P(1)%20%3D%20k$ , $P(2)%20%3D%202k$ , and $P(3)%20%3D%20k$ .
Step 2: Simplify the equation:
Combine the terms: $0.2%20%2B%204k%20%3D%201$ Subtract $0.2$ from both sides: $4k%20%3D%200.8$ Divide by $4$ to find $k$ : $k%20%3D%200.2%20%3D%20%5Cfrac%7B1%7D%7B5%7D$
Step 3: Find $P(2)$ : Given $P(2)%20%3D%202k$ , substitute the value of $k$ : $P(2)%20%3D%202%20%5Ctimes%20%5Cfrac%7B1%7D%7B5%7D%20%3D%20%5Cfrac%7B2%7D%7B5%7D$
Conclusion: The value of $P(2)$ is $%5Cmathbf%7B%5Cfrac%7B2%7D%7B5%7D%7D$ .

Step 1: Find $k$
The direction ratios of the first line are

%5Clangle%20-3%2C%202k%2C%202%20%5Crangle

and for the second line are

%5Clangle%203k%2C%201%2C%20-7%20%5Crangle

. Since the lines are perpendicular:

(-3)(3k)%20%2B%20(2k)(1)%20%2B%20(2)(-7)%20%3D%200

-9k%20%2B%202k%20-%2014%20%3D%200%20%5Cimplies%20-7k%20%3D%2014%20%5Cimplies%20k%20%3D%20-2

Step 2: Find the vector equation of the perpendicular line
The direction vectors are:

%5Cvec%7Bb%7D_1%20%3D%20%5Clangle%20-3%2C%20-4%2C%202%20%5Crangle%2C%20%5Cquad%20%5Cvec%7Bb%7D_2%20%3D%20%5Clangle%20-6%2C%201%2C%20-7%20%5Crangle

The perpendicular vector is:

%5Cvec%7Bb%7D%20%3D%20%5Cvec%7Bb%7D_1%20%5Ctimes%20%5Cvec%7Bb%7D_2%20%3D%20%5Cbegin%7Bvmatrix%7D%20%5Chat%7Bi%7D%20%26%20%5Chat%7Bj%7D%20%26%20%5Chat%7Bk%7D%20%5C%5C%20-3%20%26%20-4%20%26%202%20%5C%5C%20-6%20%26%201%20%26%20-7%20%5Cend%7Bvmatrix%7D%20%3D%2026%5Chat%7Bi%7D%20-%2033%5Chat%7Bj%7D%20-%2027%5Chat%7Bk%7D

The required equation is:

%5Cvec%7Br%7D%20%3D%20%5Clangle%203%2C%20-4%2C%207%20%5Crangle%20%2B%20%5Clambda(26%5Chat%7Bi%7D%20-%2033%5Chat%7Bj%7D%20-%2027%5Chat%7Bk%7D)

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Elementary Operation Transformation Of A Matrix

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Chapter Questions
3 MCQs