Advertisement Remove all ads

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

Advertisement Remove all ads
Advertisement Remove all ads
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"`.

Advertisement Remove all ads

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
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×