^>^6.3 Vectors In the Plane
Quantities such as force and velocity involve both magnitude and direction, and cannot be completely characterized by a single real number. To represent such a quantity, a vector is used, and is detonated by a boldface lowercase letter. The magnitude can be found by using the distance formula. Vectors with the same magnitude and direction are equivalent.
Component Form of a Vector
It is convenient to describe vectors in standard position, or component form, with the initial point at the origin, and the terminal (v₁, v₂), denoted as v = ⟨v₁, v₂⟩..
The component form of the vector with initial point P(p₁, p₂) and terminal point Q(q₁, q₂) is given by:
PQ = ⟨q₁ -p₁, q₂ -p₂⟩ = ⟨v₁, v₂⟩ = v
The magnitude of v is given by:
∥v∥ = √(v²₁ + v²₂)
If ∥v∥ = 1, then v is a unit vector, if ∥v∥ = 0 it is a zero vector.
Two vectors are equal if their component forms are the same.
It is convenient to describe vectors in standard position, or component form, with the initial point at the origin, and the terminal (v₁, v₂), denoted as v = ⟨v₁, v₂⟩..
The component form of the vector with initial point P(p₁, p₂) and terminal point Q(q₁, q₂) is given by:
PQ = ⟨q₁ -p₁, q₂ -p₂⟩ = ⟨v₁, v₂⟩ = v
The magnitude of v is given by:
∥v∥ = √(v²₁ + v²₂)
If ∥v∥ = 1, then v is a unit vector, if ∥v∥ = 0 it is a zero vector.
Two vectors are equal if their component forms are the same.
Finding the Component Form of a Vector
Vector Operations
The two basic vector operations are scalar multiplication and vector addition. Geometrically, the product of vector v and scalar k is the vector that is ∣k∣ times as long as v. If k is positive, kv has the same direction as v, if negative, then kv has the opposite direction of v.
To add two vectors u and v geometrically, first position them so that the initial point of the second vector v coincides with the terminal point of the first vector u. The sum
u + v
is the vector formed by joining the initial point of the first vector u with the terminal point of the second vector v. It is known as the parallelogram law.
u + v = 〈u₁ +v₁, u₂+ v₂〉
and the scalar multiple of k times u is the vector
ku = k〈u₁, u₂〉 = 〈ku₁, ku₂〉.
The two basic vector operations are scalar multiplication and vector addition. Geometrically, the product of vector v and scalar k is the vector that is ∣k∣ times as long as v. If k is positive, kv has the same direction as v, if negative, then kv has the opposite direction of v.
To add two vectors u and v geometrically, first position them so that the initial point of the second vector v coincides with the terminal point of the first vector u. The sum
u + v
is the vector formed by joining the initial point of the first vector u with the terminal point of the second vector v. It is known as the parallelogram law.
u + v = 〈u₁ +v₁, u₂+ v₂〉
and the scalar multiple of k times u is the vector
ku = k〈u₁, u₂〉 = 〈ku₁, ku₂〉.
Properties of Vector Addition and Scalar Multiplication
Let u, v, and w be vectors and let c and d be scalars.
1. u + v = v + u
2. (u + v) + w = u + (v + w)
3. u + 0 = u
4. u + (-u) = 0
5. c(du) = cd(u)
6. (c + d) u = cu + du
7. c(u + v) = cu + cv
8. 1(u) = u, 0(u) = 0
9. ∥cv∥ = ∣c∣ ∥v∥
Let u, v, and w be vectors and let c and d be scalars.
1. u + v = v + u
2. (u + v) + w = u + (v + w)
3. u + 0 = u
4. u + (-u) = 0
5. c(du) = cd(u)
6. (c + d) u = cu + du
7. c(u + v) = cu + cv
8. 1(u) = u, 0(u) = 0
9. ∥cv∥ = ∣c∣ ∥v∥
Unit Vector
u = unit vector = v/∥v∥
u = unit vector = v/∥v∥
Finding a Unit Vector
Finding Direction Angles of Vectors
Finding Component Form
Find the tension in each cable supporting the load.