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The two vectors j+k and 3i−j+4k represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A - Mathematics

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The two vectors `hatj+hatk " and " 3hati-hatj+4hatk` represent the two sides AB and AC, respectively of a ∆ABC. Find the length of the median through A

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Solution

In ABC,

Using the triangle law of vector addition, we have

`vec(BC)=vec(AC)-vec(AB)`

`=(3hati-hatj+4hatk)-(hatj+hatk)`

`=3hati-2hatj+3hatk`

`:.vec(BD)=1/2vec(BC)=3/2hati-hatj+3/2hatk " (Since AD is the median)"`

In ∆ABD, using the triangle law of vector addition, we have

`vec(AD)=vec(AB)+vec(BD)`

`=(hatj+hatk)+(3/2hati-hatj+3/2hatk)`

`=3/2hati+0hatj+5/2hatk`

`:.AD=sqrt((3/2)^2+0^2+(5/2)^2)=1/2sqrt34`

Hence, the length of the median through A is `1/2sqrt34`units.

Concept: Position Vector of a Point Dividing a Line Segment in a Given Ratio
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