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Trigonometry - Vectors

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    Study Guide Trigonometry – Vectors 1. The Rectangular Coordinate System Figure 1 Vectors drawn on a plane. In this chapter, we focus on vectors in a two - dimensional coordinate plane . The same ideas can be extended to higher dimensions later, but for now, we’ll keep everything in 2D to make the concepts clear. 1. 1 Vectors and Standard Position A vector 𝐴𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ represents both magnitude (length) and direction . If a vector 𝐴𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ is moved so that its starting point is at the origin , it is said to be in standard position . When a vector ( 𝑂𝑃 ⃗ ⃗ ⃗ ⃗ ⃗ ) starts at the origin and has the same direction and magnitude as ( 𝐴𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ ), it is called the standard vector for ( 𝐴𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ ). You may also hear a standard vector called a: • position vector • radius vector

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    Study Guide 1.2 Finding the Standard Vector Using Coordinates To find the standard vector for a vector drawn in the coordinate plane, we only need the coordinates of its endpoints . If: • A = (x a , y a ) • B = (x b , y b ) then the standard vector has coordinates: This works because the origin is at ( 0,0 ) . Example 1 : Finding a Standard Vector Figure 2 Drawing for Example 1.

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    Study Guide Given: A( - 2, - 7 ) and B(3, 2 ) Find the coordinates of point (P) so that 𝑂𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ = 𝐴𝐵 ⃗ ⃗ ⃗ ⃗ ⃗ . Figure 3 Components of a vector. Solution: So the standard vector is: 1.3 Algebraic Vectors and Components An algebraic vector is written as an ordered pair of real numbers: Here: • (a) is the horizontal component • (b) is the vertical component

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    Study Guide Two vectors are equal if their corresponding components are equal. If both components are zero, the vector is called the zero vector . 1.4 Magnitude of a Vector The magnitude (or length) of a vector ⟨ a, b ⟩ is found using the distance formula: Example 2 : Find the magnitude of u = ⟨ 3, - 5 ⟩ . Figure 4 Vector addition. 1 . 5 Vector Addition To add vectors, simply add their corresponding components . If:

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    Study Guide then: This matches the geometric idea of placing vectors tip - to - tail . 1 . 6 Scalar Multiplication Scalar multiplication means multiplying a vector by a number. If (q) is a scalar and v = ⟨ a, b ⟩ , then: This changes the length of the vector and may reverse its direction . Example 3 : Combining Vectors Given: v = ⟨ 8, - 2 ⟩ w = ⟨ 3, 7 ⟩ Find 5 v - 2w. 1 . 7 Unit Vectors A unit vector has a magnitude of 1 . To find a unit vector v in the same direction as a nonzero vector u:

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    Study Guide Example 4 : Find a unit vector in the direction of u = ⟨ 7, - 1 ⟩ 1 . 8 The Unit Vectors i and j Two special unit vectors are: Any vector v = ⟨ a, b ⟩ can be written as: Example 5 : Write u = ⟨ 5, 3 ⟩ using I and j . Figure 5 Drawing for Example 5.

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