1
IIT-JEE 2006
+3
-0.75
Let $$\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i + \widehat j - \widehat k.$$ A vector in the plane of $$\overrightarrow a$$ and $$\overrightarrow b$$ whose projection on $$\overrightarrow c$$ is $${1 \over {\sqrt 3 }},$$ is
A
$$4\widehat i - \widehat j + 4\widehat k$$
B
$$3\widehat i + \widehat j - 3\widehat k$$
C
$$2\widehat i + \widehat j - 2\widehat k$$
D
$$4\widehat i + \widehat j - 4\widehat k$$
2
IIT-JEE 2005 Screening
+4
-1
A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $$A,$$ $$B$$ and $$C.$$ If the centroid $$D$$ $$(x, y, z)$$ of triangle $$ABC$$ satisfies the relation $${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = k,$$ then the value $$k$$ is
A
$$3$$
B
$$1$$
C
$${1 \over 3}$$
D
$$9$$
3
IIT-JEE 2005 Screening
+4
-1
If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c$$ are three non-zero, non-coplanar vectors and
$$\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,\overrightarrow {{b_2}} = \overrightarrow b + {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,$$
$$\overrightarrow {{c_1}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_2}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$$
$$\overrightarrow {{c_3}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_4}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$$
then the set of orthogonal vectors is
A
$$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_3}} } \right)$$
B
$$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_2}} } \right)$$
C
$$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_1}} } \right)$$
D
$$\left( {\overrightarrow a ,\overrightarrow {{b_2}} ,\overrightarrow {{c_2}} } \right)$$
4
IIT-JEE 2004 Screening
+4
-1
If $$\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1$$ and $$\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,$$ then $$\overrightarrow b$$ is
A
$$\widehat i - \widehat j + \widehat k$$
B
$$2\widehat j - \widehat k$$
C
$$\widehat i$$
D
$$2\widehat i$$
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