If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b} \times\overrightarrow{c}) =\frac{(\overrightarrow{b} +\overrightarrow{c})}{\sqrt 2}$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is
- $\frac{3p}{4}$
- $\frac{p}{4}$
- $\frac{p}{2}$
- $p$
The Correct Option is A
Solution and Explanation
$ \Rightarrow \, \, \, \, \, \, (\overrightarrow{a}.\overrightarrow{c}) \overrightarrow{b}- (\overrightarrow{a} .\overrightarrow{b}) \overrightarrow{c} =\frac{1}{\sqrt 2}\overrightarrow{b} +\frac{1}{\sqrt 2}\overrightarrow{c} $
On equating the coefficient of $\overrightarrow{c}$ we get
$ \hspace20mm \overrightarrow{a}.\overrightarrow{b}= - \frac{1}{\sqrt 2} \Rightarrow \, \, \, |\overrightarrow{a}|| \overrightarrow{b}| \, cos \theta =\frac{1}{\sqrt 2}$
$\therefore$ $ \hspace15mm cos \theta = - \frac{1}{\sqrt 2} \Rightarrow \, \theta =\frac{3p}{4}$
Top Questions on Vector Algebra
- If \( \hat{a}, \hat{b} \) and \( \hat{c} \) are unit vectors such that \[ \hat{a} \cdot \hat{b} = \hat{a} \cdot \hat{c} = 0 \] and the angle between \( \hat{b} \) and \( \hat{c} \) is \( \frac{\pi}{6} \), then prove that: \[ \hat{a} = \pm 2 (\hat{b} \times \hat{c}) \]
Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that:
\[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.- If \( |\vec{a}|^2 = 1 \), \( |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 2 \), then the value of \( |\vec{a} + \vec{b}| \) is:
- Two vectors \( \vec{a} \) and \( \vec{b} \) are such that \( |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \). The angle between the two vectors is:
- What is the dot product of the vectors \( \mathbf{a} = (2, 3, 1) \) and \( \mathbf{b} = (1, -1, 4) \)?
- BITSAT - 2025
- Mathematics
- Vector Algebra
Questions Asked in JEE Advanced exam
- One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is:
- JEE Advanced - 2025
- Redox reactions
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is- JEE Advanced - 2025
- Coordinate Geometry
- Let $ \mathbb{R} $ denote the set of all real numbers. Let $ a_i, b_i \in \mathbb{R} $ for $ i \in \{1, 2, 3\} $.
Define the functions $ f: \mathbb{R} \to \mathbb{R},\ g: \mathbb{R} \to \mathbb{R},\ h: \mathbb{R} \to \mathbb{R} $ by:
$$ f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\quad g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, $$ $$ h(x) = f(x+1) - g(x+2) $$ If $ f(x) \ne g(x) $ for every $ x \in \mathbb{R} $, then the coefficient of $ x^3 $ in $ h(x) $ is:- JEE Advanced - 2025
- Polynomials
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
JEE Advanced Notification
GMC Azamgarh Admission 2025: Dates, Courses, Fees, Eligibility, SelectJuly 22, 2025Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.