Vector Algebra

More Topic from Mathematics

Question 1

If space straight a with rightwards arrow on top space equals space fraction numerator 1 over denominator square root of 10 end fraction space left parenthesis 3 straight i with hat on top space plus straight k with hat on top right parenthesis space and space straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space minus 6 straight k with hat on top right parenthesis comma space then space the space value space of space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top space straight x space straight b with rightwards arrow on top right parenthesis space straight x left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis right square bracket space is
  • -5

  • -3

  • 5

  • 3

Solution

A.

-5

left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space left parenthesis straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top right parenthesis right square bracket
equals negative space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket space left parenthesis straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top right parenthesis straight x space left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space right square bracket
equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis. straight b with rightwards arrow on top right parenthesis straight a with rightwards arrow on top minus left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top right square bracket
space equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket left parenthesis straight a with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis plus 2. straight b with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis straight a with rightwards arrow on top minus left parenthesis straight a with rightwards arrow on top. straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top. straight a with rightwards arrow on top right parenthesis straight b with rightwards arrow on top right parenthesis right square bracket
equals space minus space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis. left square bracket 0 space plus 2 straight a with rightwards arrow on top minus left parenthesis 0 space plus straight b with rightwards arrow on top right parenthesis right square bracket
equals negative left parenthesis 2 straight a with rightwards arrow on top space minus straight b with rightwards arrow on top right parenthesis. space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis
equals negative space left parenthesis 2 straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis squared space equals space minus 4 straight a with rightwards arrow on top squared space plus 4 straight a with rightwards arrow on top. straight b with rightwards arrow on top minus straight b with rightwards arrow on top squared
space equals space minus space 4 plus 0 minus 1 space equals space minus space 5
Question 2

Let space straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus straight j with hat on top minus 2 straight k with hat on top space and space straight b with rightwards arrow on top space equals straight i with hat on top space plus straight j with hat on top. space Let space straight c with rightwards arrow on top space be space straight a space vector space such space that space vertical line straight c with rightwards arrow on top minus straight a with rightwards arrow on top vertical line equals 3
vertical line left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x straight c with rightwards arrow on top vertical line space equals space 3 space and space the space angle space between space straight c with rightwards arrow on top space and space straight a with rightwards arrow on top space straight x space straight b with rightwards arrow on top space be space 30 to the power of straight o. space Then space straight a with rightwards arrow on top. straight c with rightwards arrow on top space is space
equal space to space
  • 1/8

  • 25/8

  • 2

  • 5

Solution

C.

2

straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus straight j with hat on top minus 2 straight k with hat on top comma
straight b with rightwards arrow on top space equals space straight i with hat on top space plus straight j with hat on top space and space vertical line straight a with hat on top vertical line space equals space 3
therefore comma space straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top space equals space space 2 straight i with hat on top space minus 2 straight j with hat on top plus straight k with hat on top comma
left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x straight c with rightwards arrow on top space equals space vertical line straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line space sin space 30 space straight n with hat on top
vertical line left parenthesis straight a with rightwards arrow on top space straight x space straight b with rightwards arrow on top right parenthesis space straight x space straight c with rightwards arrow on top vertical line space equals space 3. vertical line straight c with rightwards arrow on top vertical line.1 half
3 space equals space 3 vertical line straight c with rightwards arrow on top vertical line.1 half
therefore space vertical line straight c with rightwards arrow on top vertical line space equals space 2
Now colon space vertical line straight c with rightwards arrow on top minus straight a with rightwards arrow on top vertical line space equals space 3
straight c squared space plus straight a squared space minus 2 straight c with rightwards arrow on top. straight a with rightwards arrow on top space equals space 9
4 space plus space 9 space minus 2 straight a with rightwards arrow on top. straight c with rightwards arrow on top space equals space 9
straight a with rightwards arrow on top. straight c with rightwards arrow on top space equals space 2
Question 3

Let space straight a with minus on top space equals space straight i with hat on top space plus straight j with hat on top space plus straight k comma space straight b with bar on top space equals space straight i with hat on top space minus straight j with hat on top space plus 2 straight k with hat on top space and space straight c with bar on top space equals space straight x straight i with hat on top space plus space left parenthesis straight x minus 2 right parenthesis straight j with hat on top minus straight k with hat on top.
If space the space vector space straight c with bar on top space lies space in space the space plane space of space straight a with bar on top space and space straight b with bar on top comma space then space straight x space equals
  • 0

  • -1

  • -2

  • -4

Solution

C.

-2

straight a with rightwards arrow on top space equals space straight i with hat on top space plus straight j with hat on top space plus straight k with hat on top
straight b with rightwards arrow on top space equals space straight i with hat on top space minus straight j with hat on top space plus 2 straight k with hat on top
straight c with rightwards arrow on top space equals space straight x straight i with hat on top space plus left parenthesis straight x minus 2 right parenthesis straight j with hat on top space minus straight k with hat on top
open vertical bar table row straight x cell straight x minus 2 end cell cell negative 1 end cell row 1 1 1 row 1 cell negative 1 end cell 2 end table close vertical bar space equals 0
3 straight x plus 2 space minus straight x plus 2 space equals 0
2 straight x equals negative 4
straight x equals negative 2
Question 4

If space left parenthesis top enclose straight a space straight x space top enclose straight b right parenthesis space straight x space top enclose straight c space equals space top enclose straight a space straight x space left parenthesis top enclose straight b space straight x space top enclose straight c right parenthesis comma space where space top enclose straight a comma space top enclose straight b space and space top enclose straight c space are any three vectors such that top enclose straight a. top enclose straight b space not equal to 0 space top enclose straight b. top enclose straight c space not equal to 0 comma space then space straight a with bar on top space and space straight c with bar on top space are
  • inclined at an angle of π/3 between them

  • inclined at an angle of π/6 between them

  • perpendicular

  • parallel

Solution

D.

parallel

left parenthesis top enclose straight a space straight x space top enclose straight b right parenthesis space straight x top enclose straight c space equals space top enclose straight a space straight x space left parenthesis top enclose straight b space straight x top enclose straight c right parenthesis comma space top enclose straight a. top enclose straight b space not equal to 0 comma space straight b with rightwards arrow on top. straight c with rightwards arrow on top space not equal to 0
rightwards double arrow space left parenthesis top enclose straight a. top enclose straight c right parenthesis. top enclose straight b space minus left parenthesis top enclose straight b. top enclose straight c right parenthesis top enclose straight a space equals space left parenthesis top enclose straight a. top enclose straight c right parenthesis top enclose straight b space minus left parenthesis top enclose straight a. top enclose straight b right parenthesis top enclose straight c
left parenthesis top enclose straight a. top enclose straight b right parenthesis top enclose straight c space equals left parenthesis top enclose straight b. top enclose straight c right parenthesis top enclose straight a
top enclose straight a space vertical line vertical line top enclose straight c
Question 5

Let space straight a with rightwards arrow on top space equals space straight i with hat on top space minus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight x straight i with hat on top space plus straight j with hat on top space plus left parenthesis 1 minus straight x right parenthesis space straight k with hat on top space and space
straight c with rightwards arrow on top space equals space straight y straight i with hat on top space plus straight x space straight j with hat on top space plus left parenthesis 1 plus straight x minus straight y right parenthesis straight k with hat on top space. Then space left square bracket straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top right square bracket depends on
  • only y

  • only x

  • both x and y

  • neither x nor y

Solution

D.

neither x nor y

straight a with rightwards arrow on top space equals space straight i with hat on top minus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight x straight i with rightwards arrow on top space plus straight j with rightwards arrow on top space plus left parenthesis 1 minus straight x right parenthesis straight k with hat on top space and space straight c with rightwards arrow on top space equals straight y straight i with hat on top space plus straight x straight j with hat on top space plus left parenthesis 1 plus straight x minus straight y right parenthesis straight k with hat on top
left square bracket straight a with rightwards arrow on top straight b with rightwards arrow on top straight c with rightwards arrow on top right square bracket space equals space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top straight x straight c with rightwards arrow on top right parenthesis
straight b with rightwards arrow on top space straight x straight c with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row straight x 1 cell 1 minus straight x end cell row straight y straight x cell 1 plus straight x minus straight y end cell end table close vertical bar
space equals space straight i with hat on top space left parenthesis 1 plus space straight x minus straight x minus straight x squared right parenthesis minus space straight j with hat on top space left parenthesis straight x plus straight x squared minus xy minus straight y plus xy right parenthesis space plus straight k with hat on top space left parenthesis straight x squared minus straight y right parenthesis
straight a with rightwards arrow on top. left parenthesis straight b with rightwards arrow on top straight x straight c with rightwards arrow on top right parenthesis space equals space 1
which does not depend on x and y.