Advertisements
Advertisements
प्रश्न
If `vec"a"` and `vec"b"` represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal
उत्तर
Let ABCD be a parallelogram
Let `vec"AB" = vec"a"`
`vec"AC" = vec"b"`
Since ABCD is a parallelogram , we have
`vec"AB" = vec"DC"` and `vec"BC" = vec"AD"`
In triangle ABC, by triangle law
`vec"AB" + vec"BC" = vec"AC"`
`vec"BC" = vec"AC" - vec"AB"`
`vec"BC" = vec"b" - vec"a"`
∴ `vec"AD" = vec"b" - vec"a"`
`vec"DA" + vec"AB" = vec"DB"`
`- vec"AD" + vec"AB" = vec"DB"`
`-(vec"b" - vec"a") + vec"a" = vec"DB"`
`-vec"b" + vec"a" + vec"a" = vec"DB"`
`2vec"a" - vec"b" = vec"DB"`
The sides are `vec"AB" = vec"a", vec"BC" =vec"b" - vec"a"`
`vec"DC" = vec"a", vec"AD" =vec"b" - vec"a"`
Diagnals are `vec"AC" = vec"b", vec"DB" = 2vec"a" - vec"b"`
APPEARS IN
संबंधित प्रश्न
Let `vec"a"` and `vec"b"` be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are `(vec"a" + 2vec"b")/3` and `(vec"b" + 2vec"a")/3`
If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that `vec"BE" + vec"DC" = 3/2vec"BC"`
Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side
Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram
If `vec"PO" + vec"OQ" = vec"QO" + vec"OR"`, prove that the points P, Q, R are collinear
If D is the midpoint of the aide BC of a triangle ABC, prove that `vec"AB" + vec"AC" = 2vec"AD"`
If G is the centroid of a triangle ABC, prove that `vec"GA" + vec"GB" + vec"GC" = vec0`
If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then Prove that `vec"AB" + vec"AD" + vec"CB" + vec"CD" = 4vec"EF"`
The position vectors of the vertices of a triangle are `hat"i" + 2hat"j" + 3hat"k", 3hat"i" - 4hat"j" + 5hat"k"` and `-2hat"i" + 3hat"j" - 7hat"k"`. Find the perimeter of the triangle
Show that the points A(1, 1, 1), B(1, 2, 3) and C(2, – 1, 1) are vertices of an isosceles triangle
Choose the correct alternative:
The value of `vec"AB" + vec"BC" + vec"DA" + vec"CD"` is
Choose the correct alternative:
If `vec"BA" = 3hat"i" + 2hat"j" + hat"k"` and the position vector of is `hat"i" + 3hat"j" - hat"k"`, then the position vector A is
Choose the correct alternative:
If `vec"a", vec"b"` are the position vectors A and B, then which one of the following points whose position vector lies on AB, is
Choose the correct alternative:
If `vec"r" = (9vec"a" + 7vec"b")/16`, then the point P whose position vector `vec"r"` divides the line joining the points with position vectors `vec"a"` and `vec"b"` in the ratio
