Question:
If the vectors \(\overrightarrow{AB} = 3\hat{i} + 4\hat{k}\) and \(\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\) are the sides of a triangle \(ABC\), then the length of the median through \(A\) is :
If the vectors \(\overrightarrow{AB} = 3\hat{i} + 4\hat{k}\) and \(\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\) are the sides of a triangle \(ABC\), then the length of the median through \(A\) is :
Show Hint
The median from \(A\) goes to the midpoint \(D\) of \(BC\), and \(\overrightarrow{AD} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})\).
Updated On: Nov 13, 2026
- \(\sqrt{18}\)
- \(\sqrt{72}\)
- \(\sqrt{33}\)
- \(\sqrt{45}\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Let \(D\) be the midpoint of \(BC\). Then \(\overrightarrow{AD} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2}\).
\(\overrightarrow{AB} + \overrightarrow{AC} = 8\hat{i} - 2\hat{j} + 8\hat{k}\).
\(\overrightarrow{AD} = 4\hat{i} - \hat{j} + 4\hat{k}\).
\(|\overrightarrow{AD}| = \sqrt{16 + 1 + 16} = \sqrt{33}\).
Answer: (3) \(\sqrt{33}\)
\(\overrightarrow{AB} + \overrightarrow{AC} = 8\hat{i} - 2\hat{j} + 8\hat{k}\).
\(\overrightarrow{AD} = 4\hat{i} - \hat{j} + 4\hat{k}\).
\(|\overrightarrow{AD}| = \sqrt{16 + 1 + 16} = \sqrt{33}\).
Answer: (3) \(\sqrt{33}\)
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
View More Questions
Top JEE Main Geometry and Vectors Questions
- Let the arc AC of a circle subtend a right angle at the center O. If the point B on the arc AC divides the arc AC such that: \[ \frac{\text{length of arc AB}}{\text{length of arc BC}} = \frac{1}{5} \] and \[ \overrightarrow{OC} = \alpha \overrightarrow{OA} + \beta \overrightarrow{OB}, \] then \( \alpha = \sqrt{2} (\sqrt{3}-1) \beta \) is equal to:
- Let P be the foot of the perpendicular from the point \( Q(10,-3,-1) \) on the line: \[ \frac{x-3}{7} = \frac{y-2}{-1} = \frac{z+1}{-2}. \] Then the area of the right-angled triangle PQR, where R is the point \( (3,-2,1) \), is:
- Let \( \frac{\overline{z} - i}{z - i} = \frac{1}{3}, \, z \in \mathbb{C} \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0), C \) and \( (\alpha, 0) \), is 11 square units, then \( \alpha^2 \) equals:
- Let the area of a triangle \( \triangle PQR \) with vertices \( P(5, 4) \), \( Q(-2, 4) \), and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O(2, \frac{14}{5}) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to:
- Let the position vectors of the vertices A, B, and C of a tetrahedron ABCD be \( \hat{i} + 2\hat{j} + \hat{k} \), \( \hat{i} + 3\hat{j} - 2\hat{k} \), and \( 2\hat{i} + \hat{j} - \hat{k} \) respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is \( \frac{\sqrt{10}}{3} \) and the volume of the tetrahedron is \( \frac{\sqrt{805}}{6\sqrt{2}} \), then the position vector of E is:
View More Questions
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions