Advertisements
Advertisements
प्रश्न
If the position vectors of P and Q are \[\hat{i} + 3 \hat{j} - 7 \hat{k} \text{ and } 5 \text{i} - 2 \hat{j} + 4 \hat{k}\] then the cosine of the angle between \[\vec{PQ}\] and y-axis is
विकल्प
\[\frac{5}{\sqrt{162}}\]
\[\frac{4}{\sqrt{162}}\]
\[- \frac{5}{\sqrt{162}}\]
\[\frac{11}{\sqrt{162}}\]
उत्तर
\[- \frac{5}{\sqrt{162}}\]
\[\vec{PQ} = \vec{OQ} - \vec{OP} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k} - \left( \hat{i} + 3 \hat{j} - 7 \hat{k} \right) = 4 \hat{i} - 5 \hat{j} + 11 \hat{k} \]
\[\text{ The unit vector alongy-axis is } \hat{j} .\]
\[\text{ Let } \theta \text{ be the required angle } . \]
\[\cos \theta = \frac{\vec{PQ} . \hat{j}}{\left| \vec{PQ} \right|\left| \hat{j} \right|} = \frac{- 5}{\sqrt{16 + 25 + 121}\sqrt{1}} = \frac{- 5}{\sqrt{162}}\]
APPEARS IN
संबंधित प्रश्न
Answer the following as true or false:
Two collinear vectors are always equal in magnitude.
Answer the following as true or false:
Zero vector is unique.
Answer the following as true or false:
Two vectors having same magnitude are collinear.
Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.
If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that \[\vec{OA} + \vec{OB} + \vec{OC} = \vec{OD} + \vec{OE} + \vec{OF}\]
Show that the points (3, 4), (−5, 16) and (5, 1) are collinear.
If the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j}\] and \[\vec{b} = - 6 \hat{i} + m \hat{j}\] are collinear, find the value of m.
Show that the points A (1, −2, −8), B (5, 0, −2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Using vectors show that the points A (−2, 3, 5), B (7, 0, −1) C (−3, −2, −5) and D (3, 4, 7) are such that AB and CD intersect at the point P (1, 2, 3).
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are non-zero, non-coplanar vectors, prove that the following vectors are coplanar:
(1) \[5 \vec{a} + 6 \vec{b} + 7 \vec{c,} 7 \vec{a} - 8 \vec{b} + 9 \vec{c}\text{ and }3 \vec{a} + 20 \vec{b} + 5 \vec{c}\]
Prove that the following vectors are non-coplanar:
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[2 \vec{a} - \vec{b} + 3 \vec{c} , \vec{a} + \vec{b} - 2 \vec{c}\text{ and }\vec{a} + \vec{b} - 3 \vec{c}\]
If \[\vec{a}\], \[\vec{a}\], \[\vec{c}\] are non-coplanar vectors, prove that the following vectors are non-coplanar: \[\vec{a} + 2 \vec{b} + 3 \vec{c} , 2 \vec{a} + \vec{b} + 3 \vec{c}\text{ and }\vec{a} + \vec{b} + \vec{c}\]
If \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\] then \[\vec{a} =\]
If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} , \left| \vec{a} \right| = 3, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 7,\] then the angle between \[\vec{a} \text{ and } \vec{b}\] is
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible?
If the vectors `hati - 2xhatj + 3 yhatk and hati + 2xhatj - 3yhatk` are perpendicular, then the locus of (x, y) is ______.
What is the length of the longer diagonal of the parallelogram constructed on \[5 \vec{a} + 2 \vec{b} \text{ and } \vec{a} - 3 \vec{b}\] if it is given that \[\left| \vec{a} \right| = 2\sqrt{2}, \left| \vec{b} \right| = 3\] and the angle between \[\vec{a} \text{ and } \vec{b}\] is π/4?
If θ is the angle between two vectors `veca` and `vecb` then, `veca * vecb` ≥ 0, only when
If \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to
The projection of the vector \[\hat{i} + \hat{j} + \hat{k}\] along the vector of \[\hat{j}\] is
The vectors \[2 \hat{i} + 3 \hat{j} - 4 \hat{k}\] and \[a \hat{i} + \hat{b} j + c \hat{k}\] are perpendicular if
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then the greatest value of \[\sqrt{3}\left| \vec{a} + \vec{b} \right| + \left| \vec{a} - \vec{b} \right|\]
If the angle between the vectors \[x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}\] is acute, then x lies in the interval
If \[\vec{a} \text{ and } \vec{b}\] are two unit vectors inclined at an angle θ, such that \[\left| \vec{a} + \vec{b} \right| < 1,\] then
Let \[\vec{a} , \vec{b} , \vec{c}\] be three unit vectors, such that \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] =1 and \[\vec{a}\] is perpendicular to \[\vec{b}\] If \[\vec{c}\] makes angles α and β with \[\vec{a} and \vec{b}\] respectively, then cos α + cos β =
The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is
If \[\vec{a} \text{ and }\vec{b}\] be two unit vectors and θ the angle between them, then \[\vec{a} + \vec{b}\] is a unit vector if θ =
In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.
