Vectors

Definition

• A vector is a quantity that involves both magnitude and direction.
  • 55 miles per hour is a scalar (like mass, work, energy power, temperature change)
  • 55 miles per hour to the north is a vector
• Vectors obey the commutative law for addition.
  • When applied to vectors, the commutative law for addition means that if we have two vectors of the same type, like of another displacement: B = 3 miles to the east, then A + B must equal B + A. The vector sum A + B means the vector A followed by B, while the vector sum B + A means the vector B followed by A.
• Vectors can be denoted in several ways: A, A**,** and A with an arrow above it.
• Two vectors are equal if they have the same magnitude and direction.

Displacement (which is the difference between the final and initial positions) is the prototypical example of a vector: A (displacement) = 4 miles (magnitude) to the north (direction)

Physical Quantities Represented as Vectors:

• Displacement
• Velocity
• Acceleration
• Force
• Momentum
• Electricity
• Magnetic Fields

Adding Vectors Geometrically

• Place the tail (the initial point) of one vector at the tip of the other vector

• Connect the exposed tail to the expose tip.

• The vector formed is the sum of the first two. This is called the “tip-to-tail” method of vector addition.

Subtracting Vectors Geometrically

Used to subtract one vector from another.

For example, to get A - B, simply form vector -B, which is the scalar multiple (-1)B, and add it to A.

• It is essential to know that vector subtraction is not commutative: you must perform the subtraction in the order stated in the problem.

Scalar Multiplication

A vector can be multiplied by a scalar (a number), resulting in a vector.

If the original vector is A and the scalar is k, then the scalar multiple kA is as follows:

• Magnitude of k A = | k |A (magnitude of A)
• Direction of kA = the same as A if k is positive or the opposite of A if k is negative.

Remember that when you multiply a vector times a scalar of k, the vector becomes k times longer.

Standard Basis Vectors

• Two-dimensional vectors, that is, vector that lie flat in a plane, can be written in as the sum of a horizontal vector and a perpendicular vertical vector.
• The horizontal vector is always considered a scalar multiple of what’s called the horizontal basis vector (i) and the vertical vector is a scalar multiple of the vertical basis vector (j). Both of these special vectors have a magnitude of 1, and for this reason, they’re called unit vectors.
• Unit vectors are often represented by placing a hat (caret) over the vector; for example, the unit vectors i and j are sometimes denoted: î and ĵ

Direction of a Vector

The direction of a vector can be specified by the angle it makes with the positive x-axis; you can sketch the vector and use its components to determine the angle. For example, if θ denotes the angle that vector A = 3i + 4j makes with the +x-axis, then tanθ = Aᵧ / Aₓ = 4/3.

• You can solve for θ using the inverse trig function, θ = tan⁻¹(4/3) = 53.1°

Make sure your calculator is in the correct mode, radian or degree, when using trig functions.

If A makes the angle θ with the +x-axis, then its x- and y-components are Acosθ and Asinθ respectively (where A is the magnitude of A)

Generally, any vector in the plane can be written using two perpendicular component vectors.

The Dot Product

Several physical concepts (work, electric and magnetic flux) require that we multiply the magnitude of one vector by the magnitude of the component of the other vector parallel to the first. The dot product was invented specifically for this purpose.

A vector can be multiplied by a scalar to yield another vector. For instance, we can multiply the vector A by the scalar 2 to get the scalar multiple 2A. The product is a vector with twice the magnitude and, since the scalar is positive, the same direction is A.

Additionally, we can form a scalar by multiplying two vectors. In this case, the product of the vectors is called the dot product or the scalar product, since the result is a scalar.

The Cross Product

Some physical concepts (torque, angular momentum, magnetic force) require that we multiply the magnitude of one vector by the magnitude of the components of the vector perpendicular to the first. The cross-product was invented specifically for this purpose.

Summary

• A vector is a quantity that involves both magnitude and direction

• A quantity that does not involve a direction is a scalar

• Vector addition is carried out with the “tip-to-tail” method: place the tail of one vector at the tip of the other vector, then connect the exposed tail to the exposed tip.

• Changing the sign of a vector changes its direction by 180°

• Vector subtraction is carried out with the “tip-to-tail” method, but the negative second vector is added A - B = A + (-B)

• Vectors can be expressed in terms of components using the unit basis vectors to indicate direction: i indicates the +x-direction, j indicated the +y-direction, and k indicates the +z-direction.

• Vector addition: add respective components

• Vector subtraction: subtract the respective components

• Scalar multiplication: multiply each component by the scalar

• The magnitude of any vector, A, can be computed by applying the Pythagorean Theorem to the scalar components of the sector, Aᵧ* and Aₓ: A = √(Aₓ)^2 + (Aᵧ*)^2

• The direction of any vector, A, can be specified by the angle θ, it makes with the positive x-axis such than tan(θ)= Aᵧ /Aₓ

• The dot product or scalar product is a scalar that results from multiplying the magnitude of one vector with the magnitude of the component of another vector that is parallel to the first:

  • A· B = ABcos(θ)

• The cross product is a vector that has magnitude that results from multiplying the magnitude of one vector by the magnitude of the component of another vector that’s perpendicular to the first: | A x B| = ABsin(θ). The direction of the cross product is perpendicular to both vectors.

• The right-hand rule can be used to determine the direction of the cross product

  • Point your index finger in the direction of the first vector, then point your middle finger in the direction of the second vector. Your thumb now points in the direction of the cross product

• The magnitude and direction of the cross product can be found from the determinant of a 3 x 3 matrix with the unit basis vectors forming the top row, the scalar components of the first vector forming the middle row, and the scalar components of the second vector forming the bottom row of the matrix.