# If aijkandcjka¯=i^+j^+k^ and c¯=j^-k^, find aa¯ vector bb¯ satisfying abcandaba¯×b¯=c¯ and a¯.b¯=3 - Mathematics and Statistics

Sum

If bar"a" = hat"i" + hat"j" + hat"k"  "and"  bar"c" = hat"j" - hat"k", find bar"a" vector bar"b" satisfying bar"a" xx bar"b" = bar"c"  "and"  bar"a".bar"b" = 3

#### Solution

Given: bar"a" = hat"i" + hat"j" + hat"k" ,  bar"c" = hat"j" - hat"k"

Let bar"b" = "x"hat"i" + "y"hat"j" + "z"hat"k"

Then bar"a".bar"b" = 3 gives

(hat"i" + hat"j" + hat"k").("x"hat"i" + "y"hat"j" + "z"hat"k") = 3

∴ (1)(x) + (1)(y) + (1)(z) = 3

Also, x + y + z = 3     ...(1)

Also, bar"c" = bar"a" xx bar"b"

∴ hat"j" - hat"k" = |(hat"i", hat"j" , hat"k"),(1,1,1),("x","y","z")|

= ("z - y")hat"i" - ("z - x")hat"j" + ("y - x")hat"k"

= ("z - y")hat"i" + ("x - z")hat"j" + ("y - x")hat"k"

By equality of vectors,

z - y = 0     ...(2)

x - z = 1     .....(3)

y - x = - 1     ...(4)

From (2), y = z.

From (3), x = 1 + z

Substituting these values of x and y in (1), we get

1 + z + z + z = 3

∴ z = 2/3

∴ y = z = 2/3

∴ x = 1 + z = 1 + 2/3 = 5/3

∴ bar"b" = 5/3hat"i" + 2/3hat"j" + 2/3hat"k"

i.e. bar"b" = 1/3(5hat"i" + 2hat"j" + 2hat"k")

Concept: Vector Product of Vectors (Cross)
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