If \( \overrightarrow{a}, \overrightarrow{b} \) are two non-collinear vectors, then \( |\overrightarrow{b}| \overrightarrow{a} + |\overrightarrow{a}| \overrightarrow{b} \) represents
Show Hint
- a vector parallel to an angle bisector of \( \overrightarrow{a}, \overrightarrow{b} \)
- a vector along the difference of the vectors \( \overrightarrow{a}, \overrightarrow{b} \)
- a vector along \( \overrightarrow{a} + \overrightarrow{b} \)
- a vector outside the triangle having \( \overrightarrow{a}, \overrightarrow{b} \) as adjacent sides
The Correct Option is A
Approach Solution - 1
We are given the vector expression: \[ \overrightarrow{v} = |\overrightarrow{b}| \overrightarrow{a} + |\overrightarrow{a}| \overrightarrow{b} \]
Step 1: Understanding the vector sum
This expression represents a weighted sum of the vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \), where the weights are the magnitudes of the other vector. This type of vector sum is known to produce a vector that is in the direction of the angle bisector of the two vectors.
Step 2: Geometric Interpretation
- The vector \( \overrightarrow{v} \) lies in the plane formed by \( \overrightarrow{a} \) and \( \overrightarrow{b} \). - The weights assigned to \( \overrightarrow{a} \) and \( \overrightarrow{b} \) ensure that the resultant vector is directed along the angle bisector of the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
Step 3: Conclusion
Since the given expression aligns with the well-known angle bisector theorem in vector form, the vector \( \overrightarrow{v} \) is parallel to the bisector of the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \). Thus, the correct answer is: \[ \boxed{\text{a vector parallel to an angle bisector of } \overrightarrow{a}, \overrightarrow{b}.} \]
Approach Solution -2
Step 1: Understanding the Vector Expression
We are given the vector expression:
\[ \overrightarrow{v} = |\overrightarrow{b}| \, \overrightarrow{a} + |\overrightarrow{a}| \, \overrightarrow{b} \]
This is a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \), where each vector is scaled by the magnitude of the other. Such a combination is symmetric and balances the contributions of both vectors proportionally to their lengths.
Step 2: Geometric Interpretation
- The resultant vector \( \overrightarrow{v} \) lies in the same plane as \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
- Because each vector is multiplied by the magnitude of the other, the resulting vector points along the angle bisector of the two vectors.
- This aligns with the known vector identity for the direction of the angle bisector:
\[ \text{Angle bisector} \propto |\overrightarrow{b}| \, \overrightarrow{a} + |\overrightarrow{a}| \, \overrightarrow{b} \]
Step 3: Conclusion
Based on the geometric and algebraic interpretation, we conclude that the vector \( \overrightarrow{v} \) is parallel to the angle bisector of the angle between vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
Final Answer: \[ \boxed{\text{a vector parallel to an angle bisector of } \overrightarrow{a}, \overrightarrow{b}.} \]
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
Top AP EAPCET Vectors Questions
- Let \( O(0) \), \( A(\hat{i} + \hat{j} + \hat{k}) \), \( B(-2\hat{i} + 3\hat{k}) \), \( C(2\hat{i} + \hat{j}) \), and \( D(4\hat{k}) \) be the position vectors of the points \( O, A, B, C, \) and \( D \). If a line passing through \( A \) and \( B \) intersects the plane passing through \( O, C, \) and \( D \) at the point \( R \), then the position vector of \( R \) is:
- If the position vectors of points \( A, B, C, D \) are: \[ \vec{A} = 7\hat{i} - 4\hat{j} + 7\hat{k},\quad \vec{B} = \hat{i} - 6\hat{j} + 10\hat{k},\quad \vec{C} = -\hat{i} - 3\hat{j} + 4\hat{k},\quad \vec{D} = 5\hat{i} - \hat{j} + 5\hat{k} \] Then the quadrilateral \( ABCD \) is:
- In parallelogram \( ABCD \), points \( M \) and \( N \) are midpoints of sides \( BC \) and \( CD \) respectively. Then the length \( AM + AN = \) ?
- Three vectors \( \vec{a}, \vec{b}, \vec{c} \) satisfy: \[ \vec{a} + \vec{b} + \vec{c} = \vec{0},\quad |\vec{a}| = 1,\quad |\vec{b}| = 3,\quad |\vec{c}| = 4 \] Then the value of: \[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = ? \]
- Let \( \vec{OA} = -4\hat{i} + 3\hat{k} \), \( \vec{OB} = 14\hat{i} + 2\hat{j} - 5\hat{k} \), and vector \( \vec{OD} \) bisects \( \angle AOB \), with \( |\vec{OD}| = \sqrt{6} \). Then: \[ \vec{OD} = ? \]
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is