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Dot Product o...

Dot Product of Two Vectors

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Q. vec a vec b vec c are unit vectors not all co-linear such that vec a timesvec c=vec c timesvec b and vec b timesvec a=vec a timesvec c. alpha beta gamma are the angles between vec a and vec b vec b and c vec c and vec a respectively.If 4(cos alpha+cos beta+cos gamma)+t=0 then value of t is

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Q.

The three vertices of a parallelogram taken in order are $(- 1, 0), (3, 1), (2, 2)$ respectively. Find the coordinates of the fourth vertex.

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Q. Vectors

\to a, \to b

and

\to c

are of the same length and when taken pairwise, they form equal angles. If

\to a =^i +^j

and

\to b =^j +^k

, then vector

\to c

can be

$^i +^k$
$\frac{1}{3}^i - \frac{4}{3}^j - \frac{1}{3}^k$
$- \frac{1}{3}^i + \frac{4}{3}^j - \frac{1}{3}^k$
$^i + 2^j +^k$

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Q. Given three unit vector a, b, c no two of which are collinear satisfying a×(b×c)=1/2b. The angle between a and b

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Q. From a committee of

8

persons, in how many ways can we choose a chairman and a vice chairman assuming one person can not hold more than one position?

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Q.

If $α$ and $β$ are complex numbers with $| β | = 1$ , find $∣ ∣ \begin{matrix} \frac{β - α}{1 - ¯ ¯ ¯ α β} \end{matrix} ∣ ∣$ .

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Q.

Let two non-collinear unit vectors $^a and^b$ form an acute angle. A point P moves so that at any time t the position vector
$\to O P$ (where, O is the origin) is given by
$^a cos t +^b sin t$ . When P is farthest from origin O, let M be the length of $\to O P$ and $^u$ be the unit vector along $\to O P$ . Then,

$^u = \frac{^a +^b}{|^a +^b |} and M = (1 +^a .^b)^{\frac{1}{2}}$
$^u = \frac{^a -^b}{|^a -^b |} and M = (1 +^a .^b)^{\frac{1}{2}}$
$^u = \frac{^a +^b}{|^a +^b |} and M = (1 + 2^a .^b)^{\frac{1}{2}}$
$^u = \frac{^a -^b}{|^a -^b |} and M = (1 + 2^a .^b)^{\frac{1}{2}}$

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Q. 4^{†an²x}-2^{sec²c} +1=0 , x∈\lbrack0 , 20\rbrack

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Q.

Let $\to a =^i +^j +^k, \to b =^i -^j +^k$ and $\to c =^i -^j -^k$ be three vectors. A vector $\to v$ in the plane of
$\to a$ and $\to b$ , whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ , is given by

$^i - 3^j + 3^k$
$- 3^i - 3^j +^k$
$3^i -^j + 3^k$
$^i - 3^j - 3^k$

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Q. Let

\to a

and

\to b

be two unit vectors such that

\to a \cdot \to b = 0

. For some

x, y \in R

, let

\to c = x \to a + y \to b + (\to a \times \to b)

. If

| \to c | = 2

and the vector

\to c

is inclined at the same angle

α

to both

\to a

and

\to b

, then the value of

8 {cos}^{2} α

is .

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Q. Given that

\to a

and

\to b

are two unit vectors such that the angle between

\to a

and

\to b

is

{cos}^{- 1} (\frac{1}{4}) .

Let

\to c

be a vector in the plane of

\to a

and

\to b,

such that

| \to c | = 4

and

\to c \times \to b = 2 \to a \times \to b .

If

\to c = λ \to a + μ \to b,

where

λ

and

μ

are constants, then which of the following is/are CORRECT?

$λ = 2.$
Sum of all possible values of $μ$ is $- 1.$
Product of all possible values of $μ$ is $- 12.$
Number of distinct values of $μ$ is $3.$

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Q. If

\to a and \to b

are two unit vectors such that

\to a = ˆ i

and

\to b = ˆ j

then the angle between

\to a + \to b

and

\to a - \to b

is

$0^{ο}$
$90^{ο}$
$180^{ο}$
$45^{ο}$

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Q.

If \to A = 2 ˆ i + 3 ˆ j - 6 ˆ k and \to B = 3 ˆ i - 6 ˆ j - 5 ˆ k are the two adjacent sides of a parallelogram then the angle between the diagonals of the parallelogram is

${sin}^{- 1} (\frac{- 21}{\sqrt{12865}})$
${cos}^{- 1} (\frac{- 21}{\sqrt{12865}})$
$90^{ο}$
$0^{ο}$

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Q.

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of $cos (P + Q) + cos (Q + R) + cos (R + P)$ is

$- \frac{3}{2}$
$\frac{3}{2}$
$\frac{5}{3}$
$- \frac{5}{3}$

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Q. Let

\to a =^i + α^j + 3^k

and

\to b = 3^i - α^j +^k

. If the area of the parallelogram whose adjacent sides are represented by the vectors

\to a

and

\to b

is

8 \sqrt{3}

square units, then

\to a \cdot \to b

is equal to

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Q.

$T h e m a g n i t u d e o f t h e p r o j e c t i o n o f 2^i + 3^j + 4^k o n t h e v e c t o r^i +^j +^k w i l l b e - --- -$

$\sqrt{3}$
$\sqrt{\frac{3}{2}}$
$3 \sqrt{3}$
None of the above

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Q.

$I f \to A = 2^i - 3^j and \to B = - 4^i + 2^j, then | \to A . \to B | =$ ___

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Q.

Let P, Q, R and S be the points on the plane with position vectors
$- 2^i -^j, 4^i, 3^i + 3^j a n d - 3^i + 2^j$ , respectively. The quadrilateral PQRS must be a

parallelogram, which is neither a rhombus nor a rectangle
square
rectangle, but not a square
rhombus, but not a square

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