Maths-
General
Easy

Question

Let a with not stretchy bar on top equals 2 ı with not stretchy bar on top plus 3 ȷ with not stretchy bar on top plus k with not stretchy bar on top comma b with not stretchy bar on top equals 4 ı with not stretchy bar on top plus ȷ with not stretchy bar on top and c with not stretchy bar on top equals ı with not stretchy bar on top minus 3 ȷ with not stretchy bar on top minus 7 k with not stretchy bar on top A vector r with not stretchy bar on top  is such thatError converting from MathML to accessible text. Error converting from MathML to accessible text.

  1. l with not stretchy bar on top plus 3 ȷ with not stretchy bar on top plus 2 k with not stretchy bar on top
  2. l with not stretchy bar on top plus 3 ȷ with not stretchy bar on top minus 2 k with not stretchy bar on top
  3. l with not stretchy bar on top minus 2 ȷ with not stretchy bar on top plus 3 k with not stretchy bar on top
  4. l with not stretchy bar on top plus 2 ȷ with not stretchy bar on top plus 5 k with not stretchy bar on top

Hint:

There are four vectors. We are given three of the vectors. We have to find the fourth vector. We are given the dot product of fourth vector with each of the three vectors. We will assume variables for the components of the fourth vector. Then, we will find the dot product.

The correct answer is: l with not stretchy bar on top plus 3 ȷ with not stretchy bar on top minus 2 k with not stretchy bar on top


    The three vectors are as follows:
    a with rightwards arrow on top equals 2 i with hat on top plus 3 j with hat on top plus k with hat on top
    b with rightwards arrow on top equals 4 i with hat on top plus j with hat on top
    c with rightwards arrow on top equals i with hat on top minus 3 j with hat on top minus 7 k with hat on top
    Let the fourth vector be r with rightwards arrow on top equals x i with hat on top plus y j with hat on top plus z k with hat on top
    We will write the dot products now:
    r with rightwards arrow on top. a with rightwards arrow on top equals 9
r with rightwards arrow on top. b with rightwards arrow on top space equals 7
r with rightwards arrow on top. c with rightwards arrow on top space equals 6
    We will take the dot products of the given vectors with r with rightwards arrow on top.
    r with rightwards arrow on top. a with rightwards arrow on top space equals space 9
left parenthesis x i with hat on top space plus y j with hat on top space plus z k with hat on top right parenthesis. left parenthesis 2 i with hat on top plus 3 j with hat on top plus k with hat on top right parenthesis space equals 9
2 x plus 3 y plus z space equals space 9 space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

r with rightwards arrow on top. b with rightwards arrow on top space equals 7
open parentheses x i with hat on top plus y j with hat on top plus z k with hat on top close parentheses. open parentheses 4 i with hat on top plus j with hat on top close parentheses space equals 7
4 x space plus space y space equals space 7 space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

r with rightwards arrow on top. c with rightwards arrow on top equals 6
left parenthesis x i with hat on top plus y j with hat on top plus z k with hat on top right parenthesis. left parenthesis i with hat on top minus 3 j with hat on top minus 7 k with hat on top right parenthesis space equals space 6
x space minus space 3 y space minus 7 z space equals space 6 space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
    Now we have three simultaneous equations. We will solve them to get final answer.
    Step 1: Add the three equation.
    2x + 3y + z = 9
    +        4x + y           = 7
     +        x - 3y - 7z    = 6
    _________________________________
                7x + y - 6z  =  22                    ...(4)
    Step 2:  Subtract (2) from (4)
    7x + y - 6z = 22
    -          (-)4x +(-) y       = 7
    _________________________________
    3x - 6z    = 15
    Divide both the sides by 3.
    x - 2z = 5       ...(5)
    Step 3: Multiply equation (1) by 2 and add equation (5) to it.
    2(2x + 3y + z) = 2(9)
    4x + 6y + 2z  = 18
    +         x +  0  - 2z  = 5
    ______________________________
    5x + 6y = 23            ...(6)
    Step 4: Multiply equation (2) by -6 and add (6) and (2)
    -6(4x + y) = -6(7)
    -24x - 6y = -42
    +            5x + 6y =  23
    __________________________________
    -19x + 0 = -19
    x = 1
    Step 5: Substituting value of x in equation (2) and (5)
    4(1) + y = 7
    y = 3
    (1) -2z = 5
    -2z = 4
    z = -2
    Substituting the values of x,y and z we get
    r with rightwards arrow on top equals i with hat on top plus 3 j with hat on top minus 2 k with hat on top
    This is the final answer.

    We can solve simultaneous equations as we feel easy for us. There are not any particular steps to be followed. We just have to solve the equation to find the values of the variables. We try to eliminate the variables using addition and subtraction to single out single variable.

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