How to Find a Vector That Is Perpendicular

To find perpendicular vectors you can use the dot product or the cross product.
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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.

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