Maths · Coordinates of a point in space, the distance between two points, section formula
Then the ratio in which divides is
\( \operatorname{Let} \overrightarrow{\boldsymbol{A}}=\hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}+\boldsymbol{3} \hat{\boldsymbol{k}}, \overrightarrow{\boldsymbol{B}}=\boldsymbol{4} \hat{\boldsymbol{i}}+2 \hat{\boldsymbol{j}}, \overrightarrow{\boldsymbol{C}}= \) \( \mathbf{2} \hat{\mathbf{i}}+\mathbf{2} \hat{\mathbf{j}}+\mathbf{2} \hat{k} . \) Then the ratio in which \( \boldsymbol{C} \) divides \( A B \) is
- A. 3: 4
- B. 1: 3
- C. 1: 2
- D. 1: 1
Step-by-step solution
Let C divide AB in ratio λ:1 (internal). Then C = (A + λB)/(1+λ). Equating components gives 1+4λ = 2(1+λ) and 3 = 2(1+λ), both yielding λ = 0.5. Hence ratio λ:1 = 0.5:1 = 1:2, i.e., 1:2.
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