WEBVTT
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Given that π΄π΅πΆπ· is a square with side length 27 centimeters and π hat is the unit vector perpendicular to its plane, determine the cross product of ππ and ππ.
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In this question, weβre being asked to calculate the cross product.
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And we know the cross product of vector π and vector π is equal to the magnitude of vector π multiplied by the magnitude of vector π multiplied by sin of angle π multiplied by the unit vector π§.
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π is the angle between the two vectors.
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And the unit vector π§ is perpendicular to both vector π and vector π.
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We are told that the square π΄π΅πΆπ· has side length 27 centimeters.
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As the magnitude of a vector is equal to its length, the magnitude of vector ππ is equal to 27.
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We also need to calculate the magnitude of vector ππ.
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As triangle π΄π΅πΆ is a right triangle, we can do this using the Pythagorean theorem, which states that π squared plus π squared is equal to π squared, where π is the length of the longest side or hypotenuse.
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In this question, we have the magnitude of ππ squared plus the magnitude of ππ squared is equal to the magnitude of ππ squared.
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We know that the magnitude or length of ππ and ππ is 27.
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27 squared plus 27 squared is equal to 1458.
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We can then square root both sides of this equation so that the magnitude of ππ is equal to 27 root two.
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Our next step is to redraw our diagram so that the tails or start points of both vectors are at the same point.
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We need to calculate the angle π between vector ππ and vector ππ.
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As the diagonal in a square cuts a right angle in half and a half of 90 degrees is 45 degrees, then π will be equal to 180 minus 45 degrees.
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The angle between vector ππ and vector ππ is 135 degrees.
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We can now calculate the cross product of vector ππ and vector ππ.
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ππ cross ππ is equal to the magnitude of ππ multiplied by the magnitude of ππ multiplied by sin of angle π multiplied by the unit vector π, as this is the unit vector perpendicular to the plane.
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Substituting in our values, we have 27 multiplied by 27 root two multiplied by sin of 135 degrees multiplied by the unit vector π.
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sin of 135 degrees is equal to root two over two.
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We need to multiply this by 27, 27 root two, and the unit vector π.
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Root two multiplied by root two over two is equal to one.
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So we are left with 27 multiplied by 27 multiplied by the unit vector π.
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This is equal to 729π.
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The cross product of ππ and ππ is equal to 729π.