How to Find a Vector That Is Perpendicular

To find perpendicular vectors you can use the dot product or the cross product.
••• robertiez/iStock/Getty Images

To construct a vector that is perpendicular to another given vector, you can use techniques based on the dot-product and cross-product of vectors. The dot-product of the vectors A = (a1, a2, a3) and B = (b1, b2, b3) is equal to the sum of the products of the corresponding components: A∙B = a1_b2 + a2_b2 + a3_b3. If two vectors are perpendicular, then their dot-product is equal to zero. The cross-product of two vectors is defined to be A×B = (a2_b3 - a3_b2, a3_b1 - a1_b3, a1_b2 - a2*b1). The cross product of two non-parallel vectors is a vector that is perpendicular to both of them.

Two Dimensions -- Dot Product

    Write down a hypothetical, unknown vector V = (v1, v2).

    Calculate the dot-product of this vector and the given vector. If you are given U = (-3,10), then the dot product is V∙U = -3 v1 + 10 v2.

    Set the dot-product equal to 0 and solve for one unknown component in terms of the other: v2 = (3/10) v1.

    Pick any value for v1. For instance, let v1 = 1.

    Solve for v2: v2 = 0.3. The vector V = (1,0.3) is perpendicular to U = (-3,10). If you chose v1 = -1, you would get the vector V’ = (-1, -0.3), which points in the opposite direction of the first solution. These are the only two directions in the two-dimensional plane perpendicular to the given vector. You can scale the new vector to whatever magnitude you want. For instance, to make it a unit vector with magnitude 1, you would construct W = V/(magnitude of v) = V/(sqrt(10) = (1/sqrt(10), 0.3/sqrt(10).

Three Dimensions -- Dot Product

    Write down a hypothetical unknown vector V = (v1, v2, v3).

    Calculate the dot-product of this vector and the given vector. If you are given U = (10, 4, -1), then V∙U = 10 v1 + 4 v2 - v3.

    Set the dot-product equal to zero. This is the equation for a plane in three dimensions. Any vector in that plane is perpendicular to U. Any set of three numbers that satisfies 10 v1 + 4 v2 - v3 = 0 will do.

    Choose arbitrary values for v1 and v2, and solve for v3. Let v1 = 1 and v2 = 1. Then v3 = 10 + 4 = 14.

    Perform the dot-product test to show that V is perpendicular to U: By the dot-product test, the vector V = (1, 1, 14) is perpendicular to the vector U: V∙U = 10 + 4 - 14 = 0.

Three Dimensions -- Cross Product

    Choose any arbitrary vector that is not parallel to the given vector. If a vector Y is parallel to a vector X, then Y = a*X for some non-zero constant a. For simplicity, use one of the unit basis vectors, such as X = (1, 0, 0).

    Calculate the cross product of X and U, using U = (10, 4, -1): W = X×U = (0, 1, 4).

    Check that W is perpendicular to U. W∙U = 0 + 4 - 4 = 0. Using Y = (0, 1, 0) or Z = (0, 0, 1) would give different perpendicular vectors. They would all lie in the plane defined by the equation 10 v1 + 4 v2 - v3 = 0.

Related Articles

How to Find a Plane With 3 Points
How to Multiply Vectors
How to Write Equations of Perpendicular & Parallel...
How to Solve the Unknown Variable of Triangles With...
How to Find the Area of a Parallelogram With Vertices
How to Calculate a Horizontal Tangent Line
How to Find a Parallel Line
How to Calculate kPa
How to Calculate Triangle Dimensions
How to Find the Vertices of an Ellipse
How to Solve a Hexagon
How to Calculate the Total Magnitude of Displacement
Standard Form of a Line
How to Find Side Lengths of Triangles
How to Determine If Matrices Are Singular or Nonsingular
How to Calculate Torsion Constant
How to Find Equations of Tangent Lines
How to Calculate SSE

Dont Go!

We Have More Great Sciencing Articles!