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Applications Scalar Triple Product

Trending Questions

Q. Volume of parallelopiped whose coterminous edges are given by

\to u =^i +^j + λ^k, \to v =^i +^j + 3^k

and

\to w = 2^i +^j +^k

1

cu. unit. If

θ

be the angle between the edges

\to u

and

\to w

, then

cos θ

can be:

$\frac{5}{7}$
$\frac{7}{6 \sqrt{3}}$
$\frac{7}{6 \sqrt{6}}$
$\frac{5}{3 \sqrt{3}}$

View Solution

Let $a =^i + 4^j + 2^k, b = 3^i - 2^j + 7^k$ and $c = 2^i -^j + 4^k$ . Find a vector d which is perpendicular to both a and b and c.d = 15

View Solution

Q. Let the volume of tetrahedron

A B C D

81

cubic units and

G_{1}

G_{2}

G_{3}

are centroids of triangular faces

A B C, A B D

and

A C D

respectively, then volume of tetrahedron

A G_{1} G_{2} G_{3}

is -

$3$
$6$
$\frac{81}{4}$
$54$

View Solution

Q. The volume of the parallelopiped constructed on diagonals of the faces of the given rectangular parallelopiped is

m

times the volume of the given parallelopiped. Then

m

is equal to

View Solution

\frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}} = a + b \sqrt{3}

, then find the value of a and b.

$a = 11, b = - 6$
$a = 10, b = - 2$
$a = 12, b = - 3$
$a = 11, b = - 4$

View Solution

Q. Match List-I with List-II and select the correct answer using the code given below the lists

List - I	List - II
P. Volume of parallelepiped determined by vectors $\to a, \to b and \to c$ is 2. Then the volume of the parallelepiped determined by vectors $2 (\to a \times \to b), 3 (\to b \times \to c) and (\to c \times \to a)$ is	1. 100
Q. Volume of parallelepiped determined by vectors $\to a, \to b and \to c$ is 5. Then the volume of the parallelepiped determined by vectors $3 (\to a + \to b), (\to b + \to c) and 2 (\to c + \to a)$ is	2. 30
R. Area of a triangle with adjacent sides determined by vectors $\to a and \to b$ is 20. Then the area of the triangle with adjacent sides determined by vectors	3. 24
S. Area of a parallelogram with adjacent sides determined by vectors $\to a and \to b$ is 30. Then the area of the parallelogram with adjacent sides determined by vectors $(\to a + \to b) and \to a$ is	4. 60

P Q R S
3 4 1 2
P Q R S
2 3 1 4
P Q R S
1 4 3 2
P Q R S
4 2 3 1

View Solution

Q. If

\to a, \to b, \to c

be three non-coplanar uni-modular vectors each inclined with other at an angle of

60^{\circ}

, then volume of the tetrahedron whose edges are

\to a, \to b

and

\to c

$4 \sqrt{2} cubic units$
$\frac{1}{6 \sqrt{2}} cubic units$
$\frac{1}{\sqrt{2}} cubic units$
$6 \sqrt{2} cubic units$

View Solution

Q. If the two diagonals of one of the faces of a parallelopiped are

6^i + 6^k

and

4^j + 2^k

and one of the edges not containing the given diagonals is

4^j - 8^k

, then the volume of the parallelopiped is

$100$
$120$
$60$
$80$

View Solution

If $\to a$ and $\to b$ are two vectors such that $| \to a | = 3, | \to b | = 2$ and angle between $\to a$ and $\to b is \frac{π}{3},$ then the area of the triangle with adjacent sides $\to a + 2 \to b$ and $2 \to a + \to b$ in sq. units is

$3 \sqrt{3}$
$9 \sqrt{3}$
$\frac{9 \sqrt{3}}{2}$
$\frac{9}{2}$

View Solution

Q. Let

\to a = (2 + sin θ)^i + cos θ^j + sin 2 θ^k

\to b = sin (θ + \frac{2 π}{3})^i + cos (θ + \frac{2 π}{3})^j + sin (2 θ + \frac{4 π}{3})^k

and

\to c = sin (θ - \frac{2 π}{3})^i + cos (θ - \frac{2 π}{3})^j + sin (2 θ - \frac{4 π}{3})^k

be three vectors where

θ \in (0, \frac{π}{2})

. The maximum volume of the tetrahedron whose coterminous edges are given by the vectors

2 \to b \times \to c, 3 \to c \times \to a

and

\to a \times 4 \to b

View Solution

Q. If a, b, c are linearly independent, then

\frac{[2 a + b, 2 b + c, 2 c + a]}{[a, b, c]} =

9
8
7
None

View Solution

Q. Volume of parallelopiped formed by vectors

\to a \times \to b, \to b \times \to c

and

\to c \times \to a

36

cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}

$a - r, b - s, c - q, d - p$
$a - p, b - s, c - q, d - r$
$a - r, b - q, c - s, d - p$
$a - s, b - r, c - q, d - p$

View Solution

Q. The volume of the tetrahedron with vertices at (1, 2, 3), (4, 3, 2), (5, 2, 7), (6, 4, 8) is

$\frac{22}{3}$
$\frac{11}{3}$
$\frac{1}{3}$
$\frac{16}{3}$

View Solution

Q. Find

λ

and

μ

if (

^i + 3^j + 9^k) \times (3^i - λ^j + μ^k) = \to 0

View Solution

Q. The image of the point

(3, - 1, 11)

w.r.t the line

\frac{x}{^{2}} = \frac{y - 2}{3} = \frac{z - 3}{4}

View Solution

Q. If

\to a, \to b, \to c

are non coplanar vectors and

λ

is a real number, then

[\begin{matrix} λ (\to a + \to b) & λ^{2} \to b & λ \to c \end{matrix}] = [\begin{matrix} \to a & \to b + \to c & \to b \end{matrix}], λ \in R

for:

exactly one value of $λ$
no value of $λ$
exactly three values of $λ$
exactly two values of $λ$

View Solution

[b \times c c \times a a \times b] =

[a, b, c]
2 [a, b, c]
0
$[a, b, c]^{2}$

View Solution

[b \times c c \times a a \times b] =

[a, b, c]
$[a, b, c]^{2}$
0
2 [a, b, c]

View Solution

Q. Let

\to u, \to v

, and

\to w

be such that

| \to u | = 1, | \to v | = 2,

and

| \to w | = 3.

If the projection of

\to v

along

\to u

is equal to that

\to w

along

\to u

, and

\to v

and

\to w

are perpendicular to each other, then |

\to u

\to v

\to w

| equals to

2
$\sqrt{7}$
$\sqrt{14}$
$14$

View Solution

Q. If a, b, c are linearly independent, then

\frac{[2 a + b, 2 b + c, 2 c + a]}{[a, b, c]} =

9
8
7
None

View Solution

Q. If the volume of a parallelopiped, whose coterminuous edges are given by the vectors

\to a =^i +^j + n^k, \to b = 2^i + 4^j - n^k

and

\to c =^i + n^j + 3^k (n \geq 0)

, is

158

cu.units, then:

$\to a \cdot \to c = 17$
$\to b \cdot \to c = 10$
$n = 9$
$n = 7$

View Solution

Q. Match

List I

with the

List II

and select the correct answer using the code given below the lists :

\begin{matrix} List I & List II (A) & Area of a triangle with adjacent sides determined by vectors \to a and \to b is 1 . Then the area of & (P) & 6 the triangle with adjacent sides determined by (3 \to a + 4 \to b) and (\to a - 3 \to b) is (B) & Volume of parallelopiped determined by vectors \to a, \to b, \to c is \frac{1}{4} . Then the volume of the & (Q) & 9 parallelopiped determined by vectors 3 (\to a + \to b), (\to b + \to c), 4 (\to c + \to a) is (C) & Area of a parallelogram with adjacent sides determined by vectors \to a and \to b is 8 . Then the & (R) & 13 area of the parallelogram with adjacent sides determined by vectors (2 \to a - \to b) and \to b is (D) & Volume of tetrahedron determined by vectors \to a, \to b and \to c is \frac{1}{2} . Then the volume of the & (S) & 16 tetrahedron determined by vectors 2 (\to a \times \to b), 3 (\to b \times \to c) and (\to c \times \to a) is \end{matrix}

Which of the following is the only CORRECT combination?

$(A) \to (R), (B) \to (Q)$
$(A) \to (R), (B) \to (P)$
$(A) \to (S), (B) \to (P)$
$(A) \to (Q), (B) \to (R)$

View Solution

Q. Let

\to a = -^i + 4^k, \to b = 5^i + 2^k

and

\to c = - 3^i +^k .

\to a = x \to b + y \to c,

then the value of

(x, y)

$(1, 1)$
$(- 2, 2)$
$(1, 2)$
$(2, - 1)$

View Solution

Q. The volume of the parallellopiped whose coterminal edges are

2^i - 3^j + 4^k,^i + 2^j - 2^k, 3^i -^j +^k

View Solution

Q. If

\to a = a_{1}^i + a_{2}^j + a_{3}^k; \to b = b_{1}^i + b_{2}^j + b_{3}^k;

\to c = c_{1}^i + c_{2}^j + c_{3}^k

and

[3 \to a + \to b, 3 \to b + \to c, 3 \to c + \to a]

= λ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a \cdot^i & \to a \cdot^j & \to a \cdot^k \to b \cdot^i & \to b \cdot^j & \to b \cdot^k \to c \cdot^i & \to c \cdot^j & \to c \cdot^k \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

then the value of

\frac{λ}{4}

View Solution

Q. The volume of the parallellopiped whose coterminal edges are

2^i - 3^j + 4^k,^i + 2^j - 2^k, 3^i -^j +^k

View Solution

Q. Volume of parallelopiped formed by vectors

\to a \times \to b, \to b \times \to c

and

\to c \times \to a

36

cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}

$a - r, b - s, c - q, d - p$
$a - p, b - s, c - q, d - r$
$a - r, b - q, c - s, d - p$
$a - s, b - r, c - q, d - p$

View Solution

Q. Let the angle between two diagonals of a cube be

arccos \frac{1}{k}

. Find

k

View Solution

Q. Angle between the vectors

\to a = -^i + 2^j +^k

and

\to b = x^i +^j + (x + 1)^k

(B) is acute angle
(A) is obtuse angle
(C) is $90^{\circ}$
(D) depends on $x$

View Solution

[b \times c c \times a a \times b] =

[a, b, c]
2 [a, b, c]
$[a, b, c]^{2}$
\N

View Solution