Learning Outcomes: (a) Find the vector product. (b) Use the properties of vector product. (c) Find the area of a parallelogram and a triangle. At the end of the lesson, students should be able to Vector Product [Cross Product] angle between and . Let [ obtained by the expansion of the determinant as stated above] The vector product of is defined as where [ is the unit vector in the direction perpendicular to the plane containing and ] • is a vector that perpendicular to both the vectors and • The direction of can be determined by the right-hand rule. Remarks: 5.3 Vector Product C05 - SM025 Page 1
Example 1 [Example 3 in PowerPoint Notes Topic 5.3] Given and . (a) Find . Solution: & (b) Hence, find the unit vectors which are perpendicular to and . Solution: & Let be the unit vectors which are perpendicular to and . Example 2 ** Given Find the angle between the vectors and . Solution : The angle between the vectors and is . be the angle between the vectors and Example 1 & 2** C05 - SM025 Page 2
angle between and . Consider Based on in same direction. in opposite direction. Since •The scalar multiple properties of 2 parallel vectors is usually applied to question. • Scalar product properties of 2 perpendicular vector is usually applied to the question. Remarks: Example 3 ** Given If and are perpendicular to each other , find the values of . Solution : Vector Product (Parallel & Perpendicular) C05 - SM025 Page 3
Given and (a) . By using the vector product, find the value of if and are parallel. Solution : By comparing component Given and (b) By using the vector product, show that and are NOT parallel. Solution : Note that . Therefore, and are NOT parallel. [Shown] Example 4** C05 - SM025 Page 4
Consider the vectors and a non-zero scalar . (I) where is zero vector. (II) (III) (IV) (V) (VI) (VII) (VIII) is parallel to (IX) and are perpendicular to each other. Example 5* [Example 2 in PowerPoint Notes Topic 5.3] Show that . Solution : By referring to the property : where is zero vector. Properties of Vector Product C05 - SM025 Page 5
Since , Area of Parallelogram C05 - SM025 Page 6
Find a vector which is perpendicular to and (a) Hence, find the area of triangle . Given the points , and Solution : Example 6 (a) C05 - SM025 Page 7
Find the position vector of point in which is a parallelogram. (b) Hence, find the area of parallelogram . Given the points , and Solution : Based on the diagram above, The position vector of point is . Example 6 (b) C05 - SM025 Page 8