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Assume
A is located at
(2,4),B is located at
(5,1) and James is travelling from
A to
B with a constant speed of
1m//s. The point
C represents his position at some point in time during the trip.
(a) (2 marks) Calculate the location of
C after he travelled for 3 seconds.

Assume $A$ is located at $(2,4), B$ is located at $(5,1)$ and James is travelling from $A$ to $B$ with a constant speed of $1 \mathrm{~m} / \mathrm{s}$ . The point $C$ represents his position at some point in time during the trip. $\newline$ (a) ( $2$ marks) Calculate the location of $C$ after he travelled for $3$ seconds.

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Find the magnitude of a three-dimensional vector

Full solution

Q. Assume

A

is located at

(2,4), B

is located at

(5,1)

and James is travelling from

A

to

B

with a constant speed of

1 \mathrm{~m} / \mathrm{s}

. The point

C

represents his position at some point in time during the trip.

\newline

(a) (

2

marks) Calculate the location of

C

after he travelled for

3

seconds.

Calculate Vector AB: Calculate the vector AB by subtracting the coordinates of A from B. Vector $AB = B - A = (5 - 2, 1 - 4) = (3, -3)$
Find Distance Traveled: Find the distance James travels in $3$ seconds. $\newline$ Distance = Speed $*$ Time = $1$ m/s $*$ $3$ s = $3$ m
Calculate Magnitude of AB: Calculate the magnitude of vector AB. Magnitude of AB = $\sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18}$
Find Unit Vector AB: Find the unit vector in the direction of $AB$ . $\newline$ Unit vector $AB = (\frac{3}{\sqrt{18}}, -\frac{3}{\sqrt{18}}) = (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$
Calculate Displacement Vector: Calculate the displacement vector for $3$ seconds. $\newline$ Displacement = Unit vector $\mathbf{AB} \times \text{Distance} = \left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right) \times 3\,\text{m}$ $\newline$ Displacement = $\left(\frac{3}{\sqrt{2}}, -\frac{3}{\sqrt{2}}\right)$
Find Coordinates of Point C: Find the coordinates of point C by adding the displacement vector to point A. $\newline$ Point C = A + Displacement = $(2, 4) + \left(\frac{3}{\sqrt{2}}, -\frac{3}{\sqrt{2}}\right)$ $\newline$ Point C = $(2 + \frac{3}{\sqrt{2}}, 4 - \frac{3}{\sqrt{2}})$
Simplify Coordinates of C: Simplify the coordinates of point C. $\newline$ Point C = $(2 + \frac{3}{\sqrt{2}}, 4 - \frac{3}{\sqrt{2}})$ $\newline$ Point C \approx $(2 + 2.12, 4 - 2.12)$ $\newline$ Point C \approx $(4.12, 1.88)$

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