A-Level Edexcel Maths Mechanics Working with Vectors: In this question i and j are hor

In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Question 1

In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O. Two cars P and... show full transcript

Worked Solution & Example Answer:In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Step 1

Find the direction of motion of Q, giving your answer as a bearing to the nearest degree.

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Answer

To find the direction of motion of Q, we can determine the vector components of its velocity. The velocity of Q is given as $(20i - 5j)$ .

To find the angle $q$ with respect to the east (i-axis), we use the tangent function:

$\tan(q) = \frac{-5}{20}$

Calculating this gives:

$q = \tan^{-1}\left(\frac{-5}{20}\right) = \tan^{-1}\left(-0.25\right)$

This results in an angle; converting to bearing, we assume a clockwise rotation and thus:

$q = 180 + (\tan^{-1}\left(0.25\right)) = 180 + 14.036\degree \approx 194\degree$

Therefore, the direction of motion of Q is approximately 194\degree.

Step 2

Find an expression for p in terms of t.

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Answer

At time $t = 0$ , the initial position vector of P is given as 400 metres.

As P moves with a velocity of $(15i + 20j) \, m \, s^{-1}$ , the position vector at time $t$ becomes:

$p = 400 + (15i + 20j) * t = (400 + 15t)i + (20t)j$ .

Step 3

Find an expression for q in terms of t.

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Answer

For Q, starting at 800 metres and moving with a velocity of $(20i - 5j) \, m \, s^{-1}$ , its position vector at time $t$ can be expressed as:

$q = 800 + (20i - 5j) * t = (800 + 20t)i + (800 - 5t)j$ .

Step 4

Find the position vector of Q when Q is due west of P.

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Answer

For Q to be due west of P, the j-components of their position vectors must be equal, thus:

$20t = 800 - 5t$

Solving for $t$ gives:

$25t = 800 \Rightarrow t = 32 \text{ seconds}$

Now substituting $t = 32$ into the expression for Q:

$q = (800 + 20(32))i + (800 - 5(32))j = (800 + 640)i + (800 - 160)j = 1440i + 640j$

Thus, the position vector of Q when it is due west of P is $1440i + 640j$ .

In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Worked Solution & Example Answer:In this question i and j are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin O - Edexcel - A-Level Maths Mechanics - Question 1 - 2016 - Paper 1

Find the direction of motion of Q, giving your answer as a bearing to the nearest degree.

Find an expression for p in terms of t.

Find an expression for q in terms of t.

Find the position vector of Q when Q is due west of P.

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