Advertisements
Advertisements
Question
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 =
Options
(a) \[- \frac{3}{2}\]
(b) \[\frac{3}{2}\]
(c) 1
(d) −1
Solution
(d) −1
\[\text{ Given that } \vec{a} , \vec{b} \text{ and } \vec{c} are \text{ unit } vectors.\]
\[\text{ So },\left| \vec{a} \right|=1,\left| \vec{b} \right|=1and\left| \vec{c} \right|=1.\]
\[\text{ Since } \vec{a} \text{ and } \vec{b} \text{ are mutually perpendicular },\]
\[ \vec{a} . \vec{b} = 0\]
\[\text{ Now },\]
\[\left| \vec{a} + \vec{b} + \vec{c} \right| = 1\]
\[ \Rightarrow \left| \vec{a} + \vec{b} + \vec{c} \right|^2 = 1\]
\[ \Rightarrow \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + \left| \vec{c} \right|^2 + 2 \vec{a} . \vec{b} + 2 \vec{b} . \vec{c} + 2 \vec{c} . \vec{a} = 1\]
\[ \Rightarrow 1 + 1 + 1 + 2\left( 0 \right) + 2 \left| \vec{a} \right| \left| \vec{b} \right| \cos \beta + 2 \left| \vec{c} \right| \left| \vec{a} \right| \cos \alpha = 1\]
\[ \Rightarrow 3 + 2 \left( \cos \alpha + \cos \beta \right) = 1\]
\[ \Rightarrow 2 \left( \cos \alpha + \cos \beta \right) = - 2\]
\[ \Rightarrow \cos \alpha + \cos \beta = - 1\]
\[\]
APPEARS IN
RELATED QUESTIONS
Classify the following measures as scalars and vectors:
(i) 15 kg
(ii) 20 kg weight
(iii) 45°
(iv) 10 meters south-east
(v) 50 m/sec2
Answer the following as true or false:
\[\vec{a}\] and \[\vec{a}\] are collinear.
Answer the following as true or false:
Two vectors having same magnitude are collinear.
If \[\vec{a}\] is a vector and m is a scalar such that m \[\vec{a}\] = \[\vec{0}\], then what are the alternatives for m and \[\vec{a}\] ?
Five forces \[\overrightarrow{AB,} \overrightarrow { AC,} \overrightarrow{ AD,}\overrightarrow{AE}\] and \[\overrightarrow{AF}\] act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 \[\overrightarrow{AO,}\] where O is the centre of hexagon.
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.
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 coplanar:
\[2 \hat{i} - \hat{j} + \hat{k} , \hat{i} - 3 \hat{j} - 5 \hat{k} \text{ and }3 \hat{i} - 4 \hat{j} - 4 \hat{k}\]
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 \[\vec{a} \cdot \text{i} = \vec{a} \cdot \left( \hat{i} + \hat{j} \right) = \vec{a} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 1,\] 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 ______.
The vector component of \[\vec{b}\] perpendicular to \[\vec{a}\] 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 \[\vec{a}\] is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ \[\vec{a}\] is a unit vector if
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
If the vectors \[3 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} + 8 \hat{k}\] are perpendicular, then λ is equal to
The projection of the vector \[\hat{i} + \hat{j} + \hat{k}\] along the vector of \[\hat{j}\] is
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
The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is
If θ is an acute angle and the vector (sin θ) \[\text{i}\] + (cos θ) \[\hat{j}\] is perpendicular to the vector \[\hat{i} - \sqrt{3} \hat{j} ,\] then θ =
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.
