How to Find a 5-Unit Vector with Opposite Direction to 3 2 -1

Introduction to Vectors and Direction

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In mathematics and physics, vectors are quantities that have both magnitude and direction. When working with vectors, it is often necessary to find specific vectors that meet certain conditions, such as a specific magnitude or direction. In this article, we will explore how to find a vector that is 5 units long and has an opposite direction to the vector 3 2 -1.

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Understanding the Given Vector

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The vector given is 3 2 -1. This can be represented as 3hat{i} 2hat{j} - hat{k}, where hat{i}, hat{j}, hat{k} are the unit vectors in the x, y, and z directions respectively. If we want to find a vector in the opposite direction, we simply negate each component:

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-3hat{i} - 2hat{j} - hat{k}.

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Unit Vector in the Opposite Direction

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A unit vector is a vector that has a magnitude of 1 and points in a specific direction. To find the unit vector in the opposite direction of -3hat{i} - 2hat{j} - hat{k}, we first need to calculate its magnitude:

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sqrt{(-3)^2 (-2)^2 (-1)^2} sqrt{9 4 1} sqrt{14}.

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The unit vector is then given by:

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frac{-3hat{i} - 2hat{j} - hat{k}}{sqrt{14}}.

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Vector of Specific Magnitude and Direction

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Given that we want a vector with a magnitude of 5 units and the same direction as the unit vector, we can simply scale the unit vector by 5:

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5 times frac{-3hat{i} - 2hat{j} - hat{k}}{sqrt{14}} frac{5}{sqrt{14}}(-3hat{i} - 2hat{j} - hat{k}).

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This vector is frac{5}{sqrt{14}}(-3hat{i} - 2hat{j} - hat{k}), which has a magnitude of 5 and is in the opposite direction to the original vector 3 2 -1.

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Conclusion

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By understanding the components of a vector, calculating its magnitude, and scaling it appropriately, we can find a vector of a specific magnitude and direction. This process can be crucial in various fields such as physics, engineering, and computer graphics. Mastering these concepts can help you solve more complex vector problems.

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Keywords: vector direction, unit vector, vector magnitude

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