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In the triangle PQR, PQ2aQR2bPQ¯=2a¯,QR¯=2b¯. The midpoint of PR is M. Find the following vectors in terms of aa¯ and bb¯: (i) PRPR¯ (ii) PMPM¯ (iii) QMQM¯. - Mathematics and Statistics

Sum

In the triangle PQR, `bar"PQ" = bar"2a", bar"QR" = bar"2b"`. The midpoint of PR is M. Find the following vectors in terms of `bar"a"` and `bar"b"`:

(i) `bar"PR"` (ii) `bar"PM"` (iii) `bar"QM"`.

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Solution

`bar"PQ" = bar"2a" , bar"QR" = bar"2b"`

(i) `bar"PR" = bar"PQ" + bar"QR"`

`= 2bar"a" + 2barb` 

(ii) ∵ M is the midpoint of PR

∴ `bar"PM" = 1/2bar"PR" = 1/2[2bar"a" + 2bar"b"]`

`= bar"a" + bar"b"`

(iii) `bar"RM" = 1/2(bar"RP") = - 1/2 bar"PR" = - 1/2(2bar"a" + 2bar"b")`

`= - bar"a" - bar"b"`

∴ `bar"QM" = bar"QR" + bar"RM"`

`= 2bar"b" - bar"a" - bar"b"`

`= bar"b" - bar"a"`

[Note: point (i) answer in the textbook is incorrect.]

Concept: Vectors and Their Types
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