Advertisements
Advertisements
Question
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
Options
(a) (−4, 7)
(b) [−4, 7]
(c) R −[−4, 7]
(d) R −(4, 7)
Advertisements
Solution
(c) R −[−4, 7]
\[\text{ Let }\theta \text{ be the angle between } \vec{a} \text{ and } \vec{b} .\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{x^2 - 3x - 28}{\sqrt{x^2 + 3^2 + 49} \sqrt{x^2 + x^2 + 4^2}}\]
\[\text{ For }\theta \text{ to be acute },\]
\[\cos \theta > 0\]
\[ \Rightarrow x^2 - 3x - 28 > 0\]
\[ \Rightarrow \left( x - 7 \right)\left( x + 4 \right) > 0\]
\[ \Rightarrow x \in \left( - \infty , - 4 \right) \cup \left( 7, \infty \right)\]
\[ \Rightarrow x \in R - \left[ - 4, 7 \right]\]
APPEARS IN
RELATED QUESTIONS
Answer the following as true or false:
\[\vec{a}\] and \[\vec{a}\] are collinear.
Answer the following as true or false:
Zero vector is unique.
Answer the following as true or false:
Two collinear vectors having the same magnitude are equal.
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 \[\vec{a}\] ?
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 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 coplanar:
\[\hat{i} + \hat{j} + \hat{k} , 2 \hat{i} + 3 \hat{j} - \hat{k}\text{ and }- \hat{i} - 2 \hat{j} + 2 \hat{k}\]
Prove that the following vectors are non-coplanar:
Prove that the following vectors are non-coplanar:
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 \[\vec{a} =\]
The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\] is a
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
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 \[\vec{a} \text{ and } \vec{b}\] 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 =
Which of the following quantities requires both magnitude (size) and direction for its complete description?
Two cars are moving at 50 km/h toward Mumbai from different cities. Are their velocity vectors equal? Why?
What does a negative vector (-\[\vec A\]) represent?
What is a position vector?
In the graphical representation of a vector, what does the arrow length represent?
