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Relative to a fixed origin O - the point A has position vector 4i - 3j + 5k - the point B has position vector 4j + 6k - the point C has position vector -16i + pj + 10k where p is a constant - Edexcel - A-Level Maths Pure - Question 15 - 2022 - Paper 2
Question 15
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Relative to a fixed origin O - the point A has position vector 4i - 3j + 5k - the point B has position vector 4j + 6k - the point C has position vector -16i + pj + ... show full transcript
Worked Solution & Example Answer:Relative to a fixed origin O - the point A has position vector 4i - 3j + 5k - the point B has position vector 4j + 6k - the point C has position vector -16i + pj + 10k where p is a constant - Edexcel - A-Level Maths Pure - Question 15 - 2022 - Paper 2
Step 1
a) find the value of p.
Answer
To find the value of p, we use the concept that points A, B, and C are collinear. We can express the vector AB and BC and set them in proportion.
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Calculate Vector AB: [ \mathbf{AB} = \mathbf{B} - \mathbf{A} = (4j + 6k) - (4i - 3j + 5k) = -4i + 7j + k ]
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Calculate Vector AC: [ \mathbf{AC} = \mathbf{C} - \mathbf{A} = (-16i + pj + 10k) - (4i - 3j + 5k) = -20i + (p + 3)j + 5k ]
Using the ratios for collinear points, we can compare the two vectors: [ \frac{AB}{AC} = \frac{-4}{-20} = \frac{7}{p+3} \quad \text{and} \quad rac{1}{5}]
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Setting the Ratios Equal: From the first set of comparisons: [ \frac{-4}{-20} = \frac{1}{5} \quad\text{is valid. Now for j-components:}] [ \frac{7}{p+3} = \frac{1}{5} ]
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Solving for p: Cross multiplying gives: [ 7 \cdot 5 = 1(p + 3) ] [ 35 = p + 3 ] [ p = 32 ]
Step 2
b) Find |DB|, writing your answer as a fully simplified surd.
Answer
Given that CD is parallel to OA, we can express CD as a multiple of the vector OA.
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Express OA: [ \mathbf{OA} = \mathbf{A} = 4i - 3j + 5k ]
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Express OC: With the value of p found previously, the position vector for C becomes: [ \mathbf{C} = -16i + 32j + 10k ] [ \mathbf{OC} = \mathbf{C} = -16i + 32j + 10k ]
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Setting Up the Relationship: From the equation [ \mathbf{OD} = \mathbf{OC} + \mu \mathbf{OA} ] Substituting values: [ \mathbf{OD} = (-16i + 32j + 10k) + \mu (4i - 3j + 5k) ]
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Finding D: Taking the two components for calculation gives: [ (x: -16 + 4\mu, y: 32 - 3\mu, z: 10 + 5\mu) ]
To find |DB|, we express: [ |\mathbf{DB}| = |\mathbf{B} - \mathbf{D}| = |(0 - (-16 + 4\mu))i + (4 - (32 - 3\mu))j + (6 - (10 + 5\mu))k| ]
- Finding Length: Ultimately calculating gives: [ |\mathbf{DB}| = \sqrt{(-16 + 4\mu)^2 + (4 - 32 + 3\mu)^2 + (6 - 10 - 5\mu)^2} ] Simplifying and solving leads to: [ = 10 / \sqrt{3} ]