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Maths-

General

Easy

Question

$stack a with rightwards arrow on top subscript i end subscript comma i element of left curly bracket 12 comma 3 right curly bracket$ are three vectors such that $open vertical bar stack a subscript i end subscript with rightwards arrow on top close vertical bar equals square root of 15$ and $open vertical bar a with rightwards arrow on top subscript i minus a with rightwards arrow on top subscript j close vertical bar equals 6$ for $i not equal to j$ . Then the value of $open square brackets stack a subscript 1 end subscript with rightwards arrow on top stack blank a subscript 2 blank end subscript with rightwards arrow on top stack a subscript 3 end subscript with rightwards arrow on top close square brackets$ is equal to

$6 \sqrt{2}$
12
24
36

The correct answer is: 36

$O left parenthesis stack 0 with minus on top right parenthesis comma A open parentheses stack a subscript 1 end subscript with minus on top close parentheses comma B open parentheses stack a subscript 2 end subscript with minus on top close parentheses comma C open parentheses stack a subscript 3 end subscript with rightwards arrow on top close parentheses$ are vertices of a tetrahedron.

Related Questions to study

The point which is at a distance of 12 units from the point whose position vector is $8 i with ‾ on top plus 10 j with ‾ on top minus 8 k with ‾ on top$ on the line which is parallel to $stack b with ‾ on top equals$ $2 i with ‾ on top plus j with ‾ on top plus 2 k with ‾ on top$ is $alpha i with ‾ on top plus beta j with ‾ on top plus gamma k with ‾ on top$ then $alpha plus beta plus gamma equals$

The point which is at a distance of 12 units from the point whose position vector is $8 i with ‾ on top plus 10 j with ‾ on top minus 8 k with ‾ on top$ on the line which is parallel to $stack b with ‾ on top equals$ $2 i with ‾ on top plus j with ‾ on top plus 2 k with ‾ on top$ is $alpha i with ‾ on top plus beta j with ‾ on top plus gamma k with ‾ on top$ then $alpha plus beta plus gamma equals$

A unit vector $stack a with rightwards arrow on top$ in the plane of $b with rightwards arrow on top equals 2 i with ˆ on top plus j with ˆ on top & c with rightwards arrow on top equals i with ˆ on top minus j with ˆ on top plus k with ˆ on top$ is such that $a with rightwards arrow on top b with rightwards arrow on top equals a with rightwards arrow on top d with rightwards arrow on top$ where $d with rightwards arrow on top equals j with ˆ on top plus 2 k with ˆ on top$ is

A unit vector $stack a with rightwards arrow on top$ in the plane of $b with rightwards arrow on top equals 2 i with ˆ on top plus j with ˆ on top & c with rightwards arrow on top equals i with ˆ on top minus j with ˆ on top plus k with ˆ on top$ is such that $a with rightwards arrow on top b with rightwards arrow on top equals a with rightwards arrow on top d with rightwards arrow on top$ where $d with rightwards arrow on top equals j with ˆ on top plus 2 k with ˆ on top$ is

If $a with rightwards arrow on top to the power of blank equals i with ˆ on top plus j with ˆ on top comma b with rightwards arrow on top to the power of ´ equals i with ˆ on top plus j with ˆ on top plus 2 k with ˆ on top & c with rightwards arrow on top to the power of ´ equals 2 i plus j with ˆ on top minus k with ˆ on top$ . Then altitude of the parallel piped formed by the vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top$ having base formed by $stack b with rightwards arrow on top & stack c with rightwards arrow on top$ is $open parentheses stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top close$ and $a with rightwards arrow on top to the power of blank comma b with rightwards arrow on top to the power of blank$ , $stack c with rightwards arrow on top to the power of ´ end exponent$ are reciprocal system of vectors)

If $a with rightwards arrow on top to the power of blank equals i with ˆ on top plus j with ˆ on top comma b with rightwards arrow on top to the power of ´ equals i with ˆ on top plus j with ˆ on top plus 2 k with ˆ on top & c with rightwards arrow on top to the power of ´ equals 2 i plus j with ˆ on top minus k with ˆ on top$ . Then altitude of the parallel piped formed by the vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top$ having base formed by $stack b with rightwards arrow on top & stack c with rightwards arrow on top$ is $open parentheses stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top close$ and $a with rightwards arrow on top to the power of blank comma b with rightwards arrow on top to the power of blank$ , $stack c with rightwards arrow on top to the power of ´ end exponent$ are reciprocal system of vectors)

parallel

If the four faces of a tetrahedron are represented by the equations $r with minus on top times left parenthesis alpha i with minus on top plus beta j with minus on top right parenthesis equals 0 comma r with minus on top times left parenthesis beta j with minus on top plus gamma k with minus on top right parenthesis equals 0 comma r with minus on top times left parenthesis gamma k with minus on top plus alpha i with minus on top right parenthesis equals 0$ and $r with minus on top times left parenthesis alpha i with minus on top plus$ $beta j with minus on top plus gamma k with minus on top right parenthesis equals P$ then volume of the tetrahedron (in cubic units) is

If the four faces of a tetrahedron are represented by the equations $r with minus on top times left parenthesis alpha i with minus on top plus beta j with minus on top right parenthesis equals 0 comma r with minus on top times left parenthesis beta j with minus on top plus gamma k with minus on top right parenthesis equals 0 comma r with minus on top times left parenthesis gamma k with minus on top plus alpha i with minus on top right parenthesis equals 0$ and $r with minus on top times left parenthesis alpha i with minus on top plus$ $beta j with minus on top plus gamma k with minus on top right parenthesis equals P$ then volume of the tetrahedron (in cubic units) is

If $sum subscript i equals 1 end subscript superscript n end superscript stack a with rightwards arrow on top subscript i end subscript equals stack 0 with rightwards arrow on top$ where $open vertical bar stack a with rightwards arrow on top subscript i end subscript close vertical bar equals 1 for all i$ , then the value of $sum subscript text i end text less or equal than text end text j less or equal than n end subscript stack a subscript i with rightwards arrow on top times a with rightwards arrow on top subscript j$ is

If $sum subscript i equals 1 end subscript superscript n end superscript stack a with rightwards arrow on top subscript i end subscript equals stack 0 with rightwards arrow on top$ where $open vertical bar stack a with rightwards arrow on top subscript i end subscript close vertical bar equals 1 for all i$ , then the value of $sum subscript text i end text less or equal than text end text j less or equal than n end subscript stack a subscript i with rightwards arrow on top times a with rightwards arrow on top subscript j$ is

The vector $stack O A with minus on top equals negative 2 i with ˆ on top plus j with ˆ on top plus 2 k with ˆ on top$ turn by a right angle about origin $left parenthesis 0 right parenthesis$ so that it passes through $j with ˆ on top plus k with ˆ on top$ . then the vector in new position is

The vector $stack O A with minus on top equals negative 2 i with ˆ on top plus j with ˆ on top plus 2 k with ˆ on top$ turn by a right angle about origin $left parenthesis 0 right parenthesis$ so that it passes through $j with ˆ on top plus k with ˆ on top$ . then the vector in new position is

parallel

The vectors $a i with ˆ on top plus j with ˆ on top plus k with ˆ on top comma i with ˆ on top plus b j plus k with ˆ on top$ and $i with ˆ on top plus j with ˆ on top plus c k with ˆ on top$ are coplanar where a,b,c are real and distinct from 1 . Then $fraction numerator 1 plus a over denominator 1 minus a end fraction plus fraction numerator 1 plus b over denominator 1 minus b end fraction plus fraction numerator 1 plus c over denominator 1 minus c end fraction equals$

The vectors $a i with ˆ on top plus j with ˆ on top plus k with ˆ on top comma i with ˆ on top plus b j plus k with ˆ on top$ and $i with ˆ on top plus j with ˆ on top plus c k with ˆ on top$ are coplanar where a,b,c are real and distinct from 1 . Then $fraction numerator 1 plus a over denominator 1 minus a end fraction plus fraction numerator 1 plus b over denominator 1 minus b end fraction plus fraction numerator 1 plus c over denominator 1 minus c end fraction equals$

The number of integer values of $p$ for which the vectors $i with ˆ on top plus p squared j with ˆ on top plus p squared k with ˆ on top comma p squared i with ˆ on top plus j with ˆ on top plus p to the power of 4 k with ˆ on top$ and $p to the power of 4 i with ˆ on top plus p to the power of 4 j with ˆ on top plus k with ˆ on top$ are coplanar is

The number of integer values of $p$ for which the vectors $i with ˆ on top plus p squared j with ˆ on top plus p squared k with ˆ on top comma p squared i with ˆ on top plus j with ˆ on top plus p to the power of 4 k with ˆ on top$ and $p to the power of 4 i with ˆ on top plus p to the power of 4 j with ˆ on top plus k with ˆ on top$ are coplanar is

The distance of the point $3 i with ˆ on top plus 5 k with ˆ on top$ from the line parallel to vector $6 i with ˆ on top plus j with ˆ on top minus 2 k with ˆ on top$ and passing through the point $8 i with ˆ on top plus 3 space j with ˆ on top plus k with ˆ on top$ is

The distance of the point $3 i with ˆ on top plus 5 k with ˆ on top$ from the line parallel to vector $6 i with ˆ on top plus j with ˆ on top minus 2 k with ˆ on top$ and passing through the point $8 i with ˆ on top plus 3 space j with ˆ on top plus k with ˆ on top$ is

parallel

The position vector of the centre of the circle $vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3$

The position vector of the centre of the circle $vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3$

Let $a with rightwards arrow on top equals 2 i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top comma b with rightwards arrow on top equals i with rightwards arrow on top plus stack 2 j with rightwards arrow on top minus k with rightwards arrow on top comma c with rightwards arrow on top equals i with rightwards arrow on top plus j with rightwards arrow on top minus 2 k with rightwards arrow on top$ , A vector in a plane of $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ whose projection on $stack a with rightwards arrow on top$ has the magnitude $square root of fraction numerator 2 over denominator 3 end fraction end root$ is

Let $a with rightwards arrow on top equals 2 i with rightwards arrow on top minus j with rightwards arrow on top plus k with rightwards arrow on top comma b with rightwards arrow on top equals i with rightwards arrow on top plus stack 2 j with rightwards arrow on top minus k with rightwards arrow on top comma c with rightwards arrow on top equals i with rightwards arrow on top plus j with rightwards arrow on top minus 2 k with rightwards arrow on top$ , A vector in a plane of $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ whose projection on $stack a with rightwards arrow on top$ has the magnitude $square root of fraction numerator 2 over denominator 3 end fraction end root$ is

The position vector of a point $P$ is given by $r with ‾ on top equals x i with ˆ on top plus y j with ˆ on top plus z k with ˆ on top$ where $x comma y comma z element of N$ and $a with ‾ on top equals i with ˆ on top plus 2 j with ˆ on top plus k with ˆ on top$ . Then the number of possible positions of $P$ such that $stack r with ‾ on top times stack a with ‾ on top equals 20$ is

The position vector of a point $P$ is given by $r with ‾ on top equals x i with ˆ on top plus y j with ˆ on top plus z k with ˆ on top$ where $x comma y comma z element of N$ and $a with ‾ on top equals i with ˆ on top plus 2 j with ˆ on top plus k with ˆ on top$ . Then the number of possible positions of $P$ such that $stack r with ‾ on top times stack a with ‾ on top equals 20$ is

parallel

If A,B,C are three points with position vectors $i with ˆ on top plus j with ˆ on top comma i with ˆ on top minus j with ˆ on top$ and $plus q j with ˆ on top plus r k with ˆ on top$ respectively, then the points are collinear if

If A,B,C are three points with position vectors $i with ˆ on top plus j with ˆ on top comma i with ˆ on top minus j with ˆ on top$ and $plus q j with ˆ on top plus r k with ˆ on top$ respectively, then the points are collinear if

$a with ‾ on top equals i with ‾ on top plus j with ‾ on top plus k with ‾ on top comma b with ‾ on top equals i with ‾ on top minus j with ‾ on top plus k with ‾ on top$ and $c with ‾ on top equals i with ‾ on top minus j with ‾ on top minus k with ‾ on top$ be the three vectors $A$ vector $stack v with ‾ on top$ in the plane of $stack a with ‾ on top & stack b with ‾ on top$ whose projection of $stack c with ‾ on top$ is $fraction numerator 1 over denominator square root of 3 end fraction$ is given by

$a with ‾ on top equals i with ‾ on top plus j with ‾ on top plus k with ‾ on top comma b with ‾ on top equals i with ‾ on top minus j with ‾ on top plus k with ‾ on top$ and $c with ‾ on top equals i with ‾ on top minus j with ‾ on top minus k with ‾ on top$ be the three vectors $A$ vector $stack v with ‾ on top$ in the plane of $stack a with ‾ on top & stack b with ‾ on top$ whose projection of $stack c with ‾ on top$ is $fraction numerator 1 over denominator square root of 3 end fraction$ is given by

The position vector of a point $P$ is $T with minus on top equals x i with ˆ on top plus y j with ˆ on top plus z k with ˆ on top$ where $x comma y comma z element of N$ and $a with minus on top equals i with ˆ on top plus 2 j with ˆ on top plus k with ˆ on top$ . If $stack r with minus on top times stack a with minus on top equals 20$ , then the number of possible positions of $P$ is

The position vector of a point $P$ is $T with minus on top equals x i with ˆ on top plus y j with ˆ on top plus z k with ˆ on top$ where $x comma y comma z element of N$ and $a with minus on top equals i with ˆ on top plus 2 j with ˆ on top plus k with ˆ on top$ . If $stack r with minus on top times stack a with minus on top equals 20$ , then the number of possible positions of $P$ is

parallel

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