Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 4a

Chapter 4, Problem 4a

If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

Verified step by step guidance

Identify the coordinates of points P₁ and P₂: P₁ = (-2, 3) and P₂ = (-1, 5).

Recall that the vector \( \mathbf{v} \) from point P₁ to point P₂ is found by subtracting the coordinates of P₁ from P₂: \( \mathbf{v} = (x_2 - x_1, y_2 - y_1) \).

Calculate the difference in the x-coordinates: \( x_2 - x_1 = -1 - (-2) \).

Calculate the difference in the y-coordinates: \( y_2 - y_1 = 5 - 3 \).

Express the vector \( \mathbf{v} \) in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{v} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Representation in the Plane

A vector in the plane can be represented as a combination of unit vectors i and j, which point in the directions of the x-axis and y-axis respectively. This allows any vector to be expressed as v = ai + bj, where a and b are the components along the x and y axes.

Vector from Two Points

The vector from point P₁ to point P₂ is found by subtracting the coordinates of P₁ from P₂. Specifically, if P₁ = (x₁, y₁) and P₂ = (x₂, y₂), then the vector v = (x₂ - x₁, y₂ - y₁), representing the displacement from P₁ to P₂.

Component Form of a Vector

The component form expresses a vector as an ordered pair of its horizontal and vertical components. Writing v = ai + bj means the vector has a horizontal component a along i and a vertical component b along j, making it easier to perform vector operations and visualize direction and magnitude.

Recommended video:

03:55

Position Vectors & Component Form

Related Practice

Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (-1, 6), P₂ = (7, -5)

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle.cos 50° cos 20° + sin 50° sin 20°

Textbook Question

The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = -5j

Textbook Question

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = 3i + j

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.