Maths · Scalar and vector products
Let be distinct real numbers. The points with position vectors ai
Let \( a, \beta, \gamma \) be distinct real numbers. The points with position vectors ai \( +\beta j+ \) \( \gamma k, \beta i+\gamma j+a k, \gamma i+a j+\beta k \)
- A. are collinear
- B. form an equilateral triangle
- C. form an scalene triangle
- D. form a right angled triangle
Step-by-step solution
Let points P, Q, R have position vectors v1, v2, v3. Compute vectors PQ = v2 - v1, PR = v3 - v1, QR = v3 - v2. Their components are cyclic permutations of (β-α, γ-β, α-γ). The squared magnitudes are equal because they are sums of squares of the same three differences in different orders. Hence all sides are equal, so triangle is equilateral.
Related MCQs
- The points , B(2,2,1) \) and \) determine a plane. The distance of the plane from the point \) is…
- 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) …
- Consider with , B \equiv(\vec{b}) \) and , \) ff =\vec{b} \cdot \vec{b}+ \) then the angle between the medians and is…