# Express the vector bar"A" = 5hat"i" - 2hat"j" + 5hat"k" as a sum of two vectors such that one is parallel to the vector bar"b" = 3hat"i" + hat"k" and other is perpendicular to bar"b". - Mathematics and Statistics

Sum

Express the vector bar"a" = 5hat"i" - 2hat"j" + 5hat"k" as a sum of two vectors such that one is parallel to the vector bar"b" = 3hat"i" + hat"k" and other is perpendicular to bar"b".

#### Solution

Let bar"a" = bar"c" + bar"d", where bar"c" is parallel to bar"b" and bar"d" is perpendicular to bar"b".

Since, bar"c" is parallel to bar"b", bar"c" = "m"bar"b", where m is a scalar.

∴ bar"c" = "m"(3hat"i" + hat"k")

i.e. bar"c" = 3"m"hat"i" + "m"hat"k"

Let bar"d" = "x"hat"i" + "y"hat"j"+ zhat"k"

Since, bar"d" is perpendicular to bar"b" = 3hat"i" + hat"k", bar"d".bar"b" = 0

∴ ("x"hat"i" + "y"hat"j" + "z"hat"k").(3hat"i" + hat"k") = 0

∴ 3x + z = 0

∴ z = - 3x

∴ bar"d" = "x"hat"i" + "y"hat"k" - 3"x"hat"k"

Now, bar"a" = bar"c" + bar"d" gives

∴ 5hat"i" - 2hat"j" + 5hat"k" = (3"m"hat"i" + "m"hat"k") + ("x"hat"i" + "y"hat"j" - 3"x"hat"k")

= (3"m" + "x")hat"i" + "y"hat"j" + ("m" - 3"x")hat"k"

By equality of vectors

3m + x = 5          ....(1)

y = - 2

and m - 3x = 5         ......(2)

From (1) and (2)

3m + x = m - 3x

∴ 2m = - 4x

∴ m = - 2x

Substituting m = - 2x in (1), we get

∴ - 6x + x = 5

∴ - 5x = 5

∴ x = - 1

∴ m = - 2x = 2

∴ bar"c" = 6hat"i" + 2hat"k" is parallel to bar"b" and bar"d" = - hat"i" - 2hat"j" + 3hat"k" is perpendicular to bar"b"

Hence, bar"a" = bar"c" + bar"d",  "where"  bar"c" = 6hat"i" + 2hat"k" and bar"d" = - hat"i" - 2hat"j" + 3hat"k"

Concept: Vectors and Their Types
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