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In this lesson, we will learn how to use vectors and operations on them to solve problems.

Q1:

Given that β π£ = 1 7 β π π΄ and β π£ = 8 β π π΅ , find β π£ π΅ π΄ .

Q2:

The forces β πΉ = οΊ β 4 β π + 6 β π ο 1 N , β πΉ = οΊ β 9 β π + 4 β π ο 2 N , and β πΉ = οΊ β 4 β π β 3 β π ο 3 N are acting on a particle, where β π and β π are two perpendicular unit vectors. Determine the magnitude the forcesβ resultant, π , and its direction π to the nearest minute.

Q3:

A book moves on a horizontal plane after being pushed by a force of 12 N parallel to the plane. The friction force between the book and the plane is 5 N. Find the resultant of these forces, and express it in terms of β π , the unit vector in the direction of the bookβs motion.

Q4:

All the sides of rhombus π π΅ πΆ π΄ are of length 5. Suppose that s i n β π΄ π π΅ = 3 4 and that π΄ π΅ > π πΆ . Use vector multiplication to find the lengths of the two diagonals.

Q5:

Given that , , , and are four collinear points, where , determine the value of which satisfies .

Q6:

π΄ π΅ πΆ π· is a square, in which the coordinates of the points π΄ , π΅ , and πΆ are ( 1 , β 8 ) , ( 3 , β 1 0 ) , and ( 5 , β 8 ) . Use vectors to determine the coordinates of the point π· and the area of the square.

Q7:

The forces β πΉ = β 1 0 β π β 7 β π 1 , β πΉ = π β π β β π 2 , and β πΉ = 5 β π + ( π β 1 0 ) β π 3 act on a particle, where β π and β π are two perpendicular unit vectors. Given that the forcesβ resultant β π = β 1 3 β π β 3 β π , determine the values of π and π .

Q8:

Given the information in the diagram below, find the value of π such that ο π΄ π· + ο π· πΈ = π ο π΄ πΆ .

Q9:

The resultant of forces β πΉ = οΊ β 4 β π + 2 β π ο 1 N , β πΉ = οΊ 5 β π β 7 β π ο 2 N , and β πΉ = οΊ 2 β π + 9 β π ο 3 N , makes an angle π with the positive π₯ -axis. Determine π , the magnitude of the resultant, and the value of t a n π .

Q10:

Given a trapezium π΄ π΅ πΆ π· , in which π΄ π· β₯ π΅ πΆ and π΄ π· π΅ πΆ = 7 , find the value of π such that ο π΄ πΆ + ο π΅ π· = π ο π΄ π· .

Q11:

In the given figure, β ο© ο© ο© ο© β π΄ π΅ and β ο© ο© ο© ο© β πΆ π· are parallel lines; however, β ο© ο© ο© ο© ο© β π π is NOT parallel to either β ο© ο© ο© ο© β π΄ π΅ or β ο© ο© ο© ο© β πΆ π· . Given that πΈ β π΄ π΅ , πΉ β πΆ π· , and π β π π , determine whether ο« π π and ο« π π are in the same, opposite, or different directions.

Q12:

Given that β πΉ = 8 β π β 5 β π 1 , β πΉ = β 1 5 β π β 5 β π 2 , and their resultant β π = β π β π β π β π , determine the values of π and π .

Q13:

Given that β π΄ = ( β 2 , 7 ) and β π΅ = ( 3 , β 8 ) , determine the area of the parallelogram whose adjacent sides are represented by β π΄ and β π΅ .

Q14:

If β π£ = β 7 6 β π π΄ π΅ and β π£ = β 5 β π π΄ , find β π£ π΅ .

Q15:

π΄ π΅ πΆ π· is a rectangle, in which the coordinates of the points π΄ , π΅ , and πΆ are ( β 1 8 , β 2 ) , ( β 1 8 , β 3 ) , and ( β 8 , π ) , respectively. Use vectors to find the value of π and the coordinates of point π· .

Q16:

Given a triangle π΄ π΅ πΆ , in which π΄ π΅ = 7 c m , π΅ πΆ = 5 6 c m , and π β π΄ π΅ πΆ = 1 2 0 β , use vectors to determine the length of π΄ πΆ .

Q17:

Trapezium π΄ π΅ πΆ π· has vertices π΄ ( 4 , 1 4 ) , π΅ ( 4 , β 4 ) , πΆ ( β 1 2 , β 4 ) , and π· ( β 1 2 , 9 ) . Given that ο π΄ π΅ β«½ ο π· πΆ and ο π΄ π΅ β ο πΆ π΅ , find the area of that trapezium.

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