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Vector Algebra

Short Answer Type

21. The position vectors of points A. B, C. D) are

straight a with rightwards arrow on top comma space space straight b with rightwards arrow on top comma space space 2 space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top

and

straight a with rightwards arrow on top space minus space 2 space straight b with rightwards arrow on top

respectively, show that

AC with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space 3 space straight b with rightwards arrow on top space and space DB with rightwards arrow on top space equals space 3 space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top

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22. Vectors drawn from the origin to the points A, B and C are respectively

straight a with rightwards arrow on top. space straight b with rightwards arrow on top space and space 4 space straight a with rightwards arrow on top space space minus space 3 space straight b with rightwards arrow on top.

Find

AC with rightwards arrow on top space and space BC with rightwards arrow on top

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23. Give a condition that the three vectors

straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top

form the three sides of a triangle. What are the other possibilities?

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Long Answer Type

24. ABCD is a parallelogram. If $AB with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space space BC with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space$ then
(i) Show that $AC with rightwards arrow on top space equals straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top comma space space space space BD with rightwards arrow on top space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top.$
(ii) What is the geometrical significance of the relation
$open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar$ ?

Here $AB with rightwards arrow on top space equals space stack straight a comma with rightwards arrow on top space space BC with rightwards arrow on top space equals space straight b with rightwards arrow on top$
(i) In $increment ABC comma space space AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top$
In $increment ABD comma space space AB with rightwards arrow on top space plus space BD with rightwards arrow on top space equals space AD with rightwards arrow on top$
$therefore space space space space BD with rightwards arrow on top space equals space AD with rightwards arrow on top space minus space AB with rightwards arrow on top space equals space BC with rightwards arrow on top space minus space AB with rightwards arrow on top$ $open square brackets because space space space AD with rightwards arrow on top space equals space BC with rightwards arrow on top close square brackets$

$equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top$
(ii)
$open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar$
$rightwards double arrow space space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close vertical bar space as space open vertical bar negative straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top close vertical bar therefore space space space space space space space space open vertical bar AC with rightwards arrow on top close vertical bar space equals space open vertical bar BD with rightwards arrow on top close vertical bar rightwards double arrow space space space space space space space space space space AC space equals space BD space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space diagonals space of space vertical line vertical line space gm space ABCD space are space equal therefore space space space space space geometrically space open vertical bar straight a plus straight b close vertical bar space equals open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space means space ABCD space is space straight a space rectangle. space$

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Short Answer Type

25.

Let $straight u with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top comma space space straight v with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space and space straight w with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 3 space straight j with hat on top.$ Find scalars a and b such that $straight w with rightwards arrow on top space equals space straight a straight u with rightwards arrow on top space plus space straight b space straight v with rightwards arrow on top.$

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26. If P₁ , P₂, P₃, P₄ are points in a plane or space and O, the origin of vectors, show that P₄ coincides with O if an only if

stack OP subscript 1 with rightwards arrow on top space plus space stack straight P subscript 1 straight P subscript 2 with rightwards arrow on top space plus space stack straight P subscript 2 straight P subscript 3 with rightwards arrow on top space plus space stack straight P subscript 3 straight P subscript 4 with rightwards arrow on top space equals space 0 with rightwards arrow on top.

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27. PQRS is a parallelogram. If

straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top

are the position vectors of the vertices P, Q and R respectively with reference to the origin of reference O ; find the position vector of S with reference to O.

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28.

If $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are the vectors determined by two adjacent sides of a regular hexagon, find the vectors determined by the other sides taken in order.

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29.

If $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top$ be the vectors represented by the sides of a triangle, taken in order, then prove that $straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top space equals space 0 with rightwards arrow on top.$

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30.

If $AO with rightwards arrow on top space plus space OB with rightwards arrow on top space equals space BO with rightwards arrow on top space plus space OC with rightwards arrow on top comma space$ then prove that the points A, B and C are collinear.