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2.6 EXERCISES 1. A particle undergoes three successive displacements in a plane as follows; 4.0 metres southeast. 5.0 metres east and 6.0 metres in a direction \( 60^{\circ} \) North of East. Find the magnitude and direction of the resultant displacement. 2. Given that \( \vec{F}_{1}=3 \hat{i}+4 \hat{j}+5 \hat{k}, \vec{F}_{2}=2 \hat{i}+3 \hat{k} \) and \( \vec{F}_{3}=6 \hat{i}-7 \hat{j}-8 \hat{k} \). Find \( 3 \overrightarrow{\mathrm{~F}}_{1}+2 \overrightarrow{\mathrm{~F}}_{2}+3 \overrightarrow{\mathrm{~F}}_{3} \). 3. A car driver traveling northeast at \( 11.5 \mathrm{~ms}^{-1} \) notices that the wind appears to be coming from \( 60^{\circ} \) south of west. Find the velocity of the wind relative to the ground. 4. A person travelling due east at \( 4 \mathrm{~ms}^{-1} \) observes that the wind appears to blow directly from the north. When he doubles his speed the wind appears to come from the northeast. Find the velocity of the wind. 5. Determine the component of the force \( -3 \hat{i}+6 \hat{j}+2 \hat{k} \) in the direction of the vector \( -4 \hat{i}+4 \hat{j}+7 \hat{k} \) 6. For what values a are \( \vec{A}=a \hat{i}-2 \hat{j}+\hat{k} \) and \( \vec{B}=2 a \hat{i}-a \hat{j}+4 \hat{k} \); perpendicular. 7. The angular velocity of a rotating rigid body is \( \vec{\omega}=4 \hat{i}+\hat{j}-2 \hat{k} \). Find the linear velocity of a point \( \mathrm{P}(5,1,3) \) on the body relative to a point \( \mathrm{Q}(3,4,: 2) \) on the axis of rotation. 8. Find the volume of a parallelepiped whose edges are given by \( \vec{A}=2 \hat{i}+3 \hat{j}-\hat{k} \), \( \hat{B}=\hat{i}-2 \hat{j}+2 \hat{k} \) and \( \hat{C}=3 \hat{i}-2 \vec{j}-\hat{k} \). 9. Find the constant, a such that the vectors \( 2 \hat{i}+\hat{j}-\hat{k}, \hat{i}+2 \hat{j}-3 \hat{k} \) and \( 2 \hat{i}+\mathrm{aj}+5 \hat{k} \) are coplanar. 10. Find the area of a triangle with vertices \( (3,-1,2),(1,-1,3) \) and \( (3,-3,1) \). 11. An airplane carrer is steaming at \( 48 \mathrm{~km} / \mathrm{h} \) southeast when a plane takes from its deck and heads in a direction which is \( 15^{\circ} \) west of north. If the speed of the plane is \( 240 \mathrm{~km} / \mathrm{h} \) what will be its displacement relative to the carrier after 2 hours? 12. Determine the resultant of accelerations of \( 18 \mathrm{~ms}^{-2} \) and \( 2.6 \mathrm{~ms}^{-2} \) inclined at an angle of \( 53^{\circ} \), and find its inclination with the smaller acceleration.

          2.6 EXERCISES
1. A particle undergoes three successive displacements in a plane as follows; 4.0 metres southeast. 5.0 metres east and 6.0 metres in a direction \( 60^{\circ} \) North of East. Find the magnitude and direction of the resultant displacement.
2. Given that \( \vec{F}_{1}=3 \hat{i}+4 \hat{j}+5 \hat{k}, \vec{F}_{2}=2 \hat{i}+3 \hat{k} \) and \( \vec{F}_{3}=6 \hat{i}-7 \hat{j}-8 \hat{k} \). Find \( 3 \overrightarrow{\mathrm{~F}}_{1}+2 \overrightarrow{\mathrm{~F}}_{2}+3 \overrightarrow{\mathrm{~F}}_{3} \).
3. A car driver traveling northeast at \( 11.5 \mathrm{~ms}^{-1} \) notices that the wind appears to be coming from \( 60^{\circ} \) south of west. Find the velocity of the wind relative to the ground.
4. A person travelling due east at \( 4 \mathrm{~ms}^{-1} \) observes that the wind appears to blow directly from the north. When he doubles his speed the wind appears to come from the northeast. Find the velocity of the wind.
5. Determine the component of the force \( -3 \hat{i}+6 \hat{j}+2 \hat{k} \) in the direction of the vector \( -4 \hat{i}+4 \hat{j}+7 \hat{k} \)
6. For what values a are \( \vec{A}=a \hat{i}-2 \hat{j}+\hat{k} \) and \( \vec{B}=2 a \hat{i}-a \hat{j}+4 \hat{k} \); perpendicular.
7. The angular velocity of a rotating rigid body is \( \vec{\omega}=4 \hat{i}+\hat{j}-2 \hat{k} \). Find the linear velocity of a point \( \mathrm{P}(5,1,3) \) on the body relative to a point \( \mathrm{Q}(3,4,: 2) \) on the axis of rotation.
8. Find the volume of a parallelepiped whose edges are given by \( \vec{A}=2 \hat{i}+3 \hat{j}-\hat{k} \), \( \hat{B}=\hat{i}-2 \hat{j}+2 \hat{k} \) and \( \hat{C}=3 \hat{i}-2 \vec{j}-\hat{k} \).
9. Find the constant, a such that the vectors \( 2 \hat{i}+\hat{j}-\hat{k}, \hat{i}+2 \hat{j}-3 \hat{k} \) and \( 2 \hat{i}+\mathrm{aj}+5 \hat{k} \) are coplanar.
10. Find the area of a triangle with vertices \( (3,-1,2),(1,-1,3) \) and \( (3,-3,1) \).
11. An airplane carrer is steaming at \( 48 \mathrm{~km} / \mathrm{h} \) southeast when a plane takes from its deck and heads in a direction which is \( 15^{\circ} \) west of north. If the speed of the plane is \( 240 \mathrm{~km} / \mathrm{h} \) what will be its displacement relative to the carrier after 2 hours?
12. Determine the resultant of accelerations of \( 18 \mathrm{~ms}^{-2} \) and \( 2.6 \mathrm{~ms}^{-2} \) inclined at an angle of \( 53^{\circ} \), and find its inclination with the smaller acceleration.

2.6 EXERCISES
1. A particle undergoes three successive displacements in a plane as follows; 4.0 metres southeast. 5.0 metres east and 6.0 metres in a direction 60^∘ North of East. Find the magnitude and direction of the resultant displacement.
2. Given that F⃗1=3 î+4 ĵ+5 k̂, F⃗2=2 î+3 k̂ and F⃗3=6 î-7 ĵ-8 k̂. Find 3 F1+2 F2+3 F3.
3. A car driver traveling northeast at 11.5 ms^-1 notices that the wind appears to be coming from 60^∘ south of west. Find the velocity of the wind relative to the ground.
4. A person travelling due east at 4 ms^-1 observes that the wind appears to blow directly from the north. When he doubles his speed the wind appears to come from the northeast. Find the velocity of the wind.
5. Determine the component of the force -3 î+6 ĵ+2 k̂ in the direction of the vector -4 î+4 ĵ+7 k̂
6. For what values a are A⃗=a î-2 ĵ+k̂ and B⃗=2 a î-a ĵ+4 k̂; perpendicular.
7. The angular velocity of a rotating rigid body is ω⃗=4 î+ĵ-2 k̂. Find the linear velocity of a point P(5,1,3) on the body relative to a point Q(3,4,: 2) on the axis of rotation.
8. Find the volume of a parallelepiped whose edges are given by A⃗=2 î+3 ĵ-k̂, B̂=î-2 ĵ+2 k̂ and Ĉ=3 î-2 j⃗-k̂.
9. Find the constant, a such that the vectors 2 î+ĵ-k̂, î+2 ĵ-3 k̂ and 2 î+aj+5 k̂ are coplanar.
10. Find the area of a triangle with vertices (3,-1,2),(1,-1,3) and (3,-3,1).
11. An airplane carrer is steaming at 48 km / h southeast when a plane takes from its deck and heads in a direction which is 15^∘ west of north. If the speed of the plane is 240 km / h what will be its displacement relative to the carrier after 2 hours?
12. Determine the resultant of accelerations of 18 ms^-2 and 2.6 ms^-2 inclined at an angle of 53^∘, and find its inclination with the smaller acceleration.

Added by Lisa B.

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