Maths · Scalar and vector products
Consider with , B \equiv(\vec{b}) \) and , \) ff =\vec{b} \cdot \vec{b}+ \) then
Consider \( \Delta A B C \) with \( A \equiv(\vec{a}), B \equiv(\vec{b}) \) and \( C=(\vec{c}), \) ff \( \vec{b} .(\vec{a}+\vec{c})=\vec{b} \cdot \vec{b}+ \) \( \vec{a} \cdot \vec{c} ;|\vec{b}-\vec{a}|=3 ;|\vec{c}-\vec{b}|=4 \) then the angle between the medians \( \overline{A M} \) and \( B D \) is
- A. \( \pi-\cos ^{-1}\left(\frac{1}{5 \sqrt{13}}\right) \)
- B. \( \pi-\cos ^{-1}\left(\frac{1}{13 \sqrt{5}}\right) \)
- C. \( \cos ^{-1}\left(\frac{1}{5 \sqrt{13}}\right) \)
- D. \( \cos ^{-1}\left(\frac{1}{13 \sqrt{5}}\right) \)
Step-by-step solution
Given conditions imply triangle ABC is a 3-4-5 right triangle with right angle at B. Using coordinates B=(0,0), A=(3,0), C=(0,4), medians AM and BD have vectors (-3,2) and (1.5,2). Dot product = -0.5, magnitudes √13 and 2.5, giving cosθ = -1/(5√13). Thus θ = π - cos⁻¹(1/(5√13)).
Related MCQs
- The vectors and are so placed that the end point of one vector is the starting point of the next vector. Then the vectors are :…
- In a triangle right angled at the vertex if the position vectors and are respectively and then the point \) lies on a line…
- Given and Find the value of \cdot\left(\boldsymbol{3} \overrightarrow{\boldsymbol{A}}_{1}-\boldsymbol{4} \boldsymbol{\vec { A }}_{2}\right) …
- Given that Out of three vectors, two are equal in magnitude and the magnitude of third vector is times that of either of the two having equa…
- The points , B(2,2,1) \) and \) determine a plane. The distance of the plane from the point \) is…