Q&A for How to Find the Angle Between Two Vectors

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How do I find the angle between two vectors? For example, vector A = 4i + 2j - 2k and vector B = 3i +2j + 3k?

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This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

wikiHow Staff Editor

Staff Answer

Use the formula with the dot product, θ = cos^-1 (a * b) / ||a|| * ||b||. To get the dot product, multiply Ai by Bi, Aj by Bj, and Ak by Bk then add the values together. To find the magnitude of A and B, use the Pythagorean Theorem (√(i^2 + j^2 + k^2). Then, use your calculator to take the inverse cosine of the dot product divided by the magnitudes and get the angle.

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If the cosine formula gives me 0, it means that the vector are perpendicular. But how do I know if it's 90° or -90° ?

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This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

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The angle between 2 vectors is always between 0° and 180°, so the angle is 90°.

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In the above example cosθ was 1/√2. But here cosθ can be 45 degrees or 315 degrees. Why is that the answer is not 315?

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This answer was written by one of our trained team of researchers who validated it for accuracy and comprehensiveness.

wikiHow Staff Editor

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When you find the angle between 2 vectors, the angle is always going to be between 0° and 180°.

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If |A + B| = |A| + |B|, then what is the angle between A and B?

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Think of the geometric representation of a vector sum. When two vectors are summed they create a new vector by placing the start point of one vector at the end point of the other (write the two vectors on paper). Now, imagine if vectors A and B both where horizontal and added. They would create a vector with the length of their two lengths added! Hence the solution is zero degrees.

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How do I find the angle between two vectors if they have the same magnitude?

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It depends on their direction. You can't call them vectors without defining their direction.

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Can you help me solve this problem? "Position vector of the point P and Q relative to the origin O are 2i and 3i+4j respectively. Find the angles between vector OP and OQ."

Community Answer

An easier way to find the angle between two vectors is the dot product formula(A.B=|A|x|B|xcos(X)) let vector A be 2i and vector be 3i+4j. As per your question, X is the angle between vectors so: A.B = |A|x|B|x cos(X) = 2i.(3i+4j) = 3x2 =6 |A|x|B|=|2i|x|3i+4j| = 2 x 5 = 10 X = cos-1(A.B/|A|x|B|) X = cos-1(6/10) = 53.13 deg The angle can be 53.13 or 360-53.13 = 306.87.

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Why can I not use cross products to find the angles?

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You can use cross products to find the angles, but then you would get the answers in terms of sine.

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Is there any way to find the angle between vectors other than dot product?

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You can use cross product or the cosine formula to determine the angles between the two vectors.

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How do I find the angle between perpendicular vectors?

Community Answer

"Perpendicular" means the angle between the two vectors is 90 degrees. To determine whether the two vectors are perpendicular or not, take the cross product of them; if the cross product is equal to zero, the vectors are perpendicular.

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If two or more angles are given with respect to the x-axis or y-axis, how can I find the magnitude?

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To find the magnitude of more than two vectors, instead of using the triangle you can use the polygon law to get the answer.

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How can I calculate a unit vector of a given vector?

Community Answer

Presumably, you are asking how to normalize the vector so its magnitude is 1.0. To do that, work out the square root of the sum of the squares of the elements. Then, divide each element by this amount. What you are doing is scaling the vector so that the sum of the squares equals 1.

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How can I find the angle between vectors who make a dot product of zero?

Siphilisiwe Munzvengi

Community Answer

If the dot product is zero, that simply means they are perpendicular; therefore, the angle is 90.

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How do you find angle between two planes defined by say ; 4x-3y+2z and 5x+2y-6z?

Community Answer

To find the angle between two non-parallel planes, you have to compute the angle between their corresponding normal vectors. By the way, the examples of plane equations you gave are not complete.

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What is the angle between two vectors in the form of a tangent?

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The tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

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