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The Cross Product

Calculus 2 · Chapter 10: Vectors & the Geometry of SpaceSection 10.3 · lesson 53 of 549 min · free

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In this lesson

Two vectors in, a perpendicular vector out.

  • Compute u\times v from a 3\times 3 determinant
  • Fix its direction with the right-hand rule
  • Read its magnitude as a parallelogram area
  • Use the scalar triple product for volume

Transcript

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MAT-265 - Calculus 2 · Section 10.3

The Cross Product

The dot product returned a scalar. A single number. The cross product does something different. It returns a vector. And not just any vector — one perpendicular to both of the inputs. That makes it the natural tool wherever a direction has to be produced from two others. Torque. Angular momentum. Surface normals in computer graphics. One important restriction. The cross product is defined only in three dimensions. Because only in three dimensions does a plane have a unique perpendicular direction.

The Definition

A Determinant Does the Bookkeeping

The definition is packaged as a three by three determinant. Put i, j and k across the top row. The components of u in the second row. And the components of v in the third. Then expand along the top row. For the i component, delete the row and column containing i. What's left is a two by two determinant from the second and third columns. For the j component, delete the row and column containing j. And here's the critical detail — the j component carries a minus sign. That's the alternating pattern of cofactor expansion. And it is the single most common source of error in this section. For the k component, delete the row and column containing k. Now — a valuable check is available at no extra cost. If the result is genuinely perpendicular to both original vectors, then each dot product must vanish. That verification takes seconds. And it catches nearly every sign error. So do it every time.

Worked Example 1

Computing a Cross Product

Let's compute one. u equals two, three, minus one. And v equals four, minus one, two. Arrange the components in the determinant. The i component. Delete the i row and column. You get three times two, minus minus one times minus one. Six minus one. Five. The j component. Delete its row and column, leaving the first and third columns. And remember the minus sign. Minus the quantity, two times two, minus minus one times four. That's minus the quantity four plus four. Minus eight. The k component. Delete its row and column, leaving the first and second. Two times minus one, minus three times four. Minus two minus twelve. Minus fourteen. Assembling: u cross v equals five, minus eight, minus fourteen. Now check it. u dotted with the result: two times five, plus three times minus eight, plus minus one times minus fourteen. Ten, minus twenty-four, plus fourteen. Zero. And v dotted with the result: twenty, plus eight, minus twenty-eight. Also zero. Both vanish. The computation is confirmed.

Direction

The Right-Hand Rule

That example confirmed the result is perpendicular to both inputs. But that condition alone does not determine the vector. Any line perpendicular to a plane admits two opposite directions. Up, or down. The choice between them is settled by a physical convention. The right-hand rule. Point the fingers of your right hand along the first vector, u. Curl them toward the second vector, v — sweeping through the smaller of the two angles between them. Your extended thumb then points in the direction of u cross v. Now here's an immediate consequence. Because reversing the order reverses the required curl, v cross u equals minus u cross v. The cross product is anticommutative. It is not commutative. Order matters. And reversing it flips the result to point the opposite way. Equal magnitude, opposite direction. One further consequence deserves note. If u and v are parallel, they lie along a single line, and span no plane at all. So no perpendicular direction is singled out. In that case the determinant has two proportional rows, and evaluates to the zero vector. And that's exactly the behavior a wrench demands. Pushing directly along the handle produces no turning effect at all.

Magnitude

The Area of a Parallelogram

The length of u cross v carries geometric meaning just as direct as its direction. Place u and v with their tails at a common point. The two vectors span a parallelogram. And elementary geometry gives its area as base times height. Take the magnitude of u as the base. The height is the perpendicular distance from the tip of v to the line containing u. Which equals the magnitude of v, times sine theta. So the magnitude of u cross v equals the magnitude of u, times the magnitude of v, times sine theta. Which is exactly the area of the parallelogram. Now compare that with the dot product, which carried cosine theta. The dot product is largest when the vectors align. The cross product is largest when they're perpendicular. The two measure complementary things. Here's an application. Find the area of a triangle from three points in space. Form two vectors from one vertex to the other two. Take their cross product. Say you get six, minus six, six. Its magnitude is root of thirty-six plus thirty-six plus thirty-six. Root one hundred eight. Simplify: root of thirty-six times three. Six root three. That's the parallelogram's area. And the triangle is exactly half of it. Three root three. About five point one nine six square units. Notice what we never did. No angle was measured. No height was constructed. The three coordinates alone determined the area. Which is precisely why this method is standard in surveying and in computer graphics.

Volume

The Scalar Triple Product

Three vectors that do not all lie in one plane span a solid called a parallelepiped. A slanted box, whose six faces are parallelograms. Its volume follows from the same base-times-height reasoning used for area. Now applied one dimension higher. Take the parallelogram spanned by b and c as the base. So the base area is the magnitude of b cross c. The height is the perpendicular distance from the tip of a, down to the base plane. And since b cross c points perpendicular to that plane, the height is the length of the projection of a onto that direction. Multiplying base by height produces the dot product of a, with b cross c. That quantity is called the scalar triple product. And it can be computed as a single three by three determinant, with the three vectors as rows. The volume is its absolute value. The absolute value is required, because the determinant may be negative — depending on the orientation of the three vectors. While a volume never can be. And there's one more use, which is arguably more important. When the triple product equals zero, the three vectors are coplanar. They lie in a common plane, and the solid collapses to a flat figure of zero volume. That's the standard test for coplanarity.

Key Takeaways

What to Carry Forward

Let's compare the two products directly. The dot product returns a scalar. The cross product returns a vector. The dot product works in any dimension. The cross product is three dimensions only. The dot product carries cosine theta. The cross product carries sine theta. The dot product is largest when the vectors align. The cross product when they're perpendicular. The dot product is zero when they're perpendicular. The cross product is zero when they're parallel. And the dot product is commutative, while the cross product is anticommutative. Complementary in almost every respect. Five takeaways. The cross product is a three by three determinant with i, j and k on the top row. And the j component carries a minus sign. Verify by dotting the result with both inputs. Both must give zero. It takes seconds and catches nearly every sign error. The right-hand rule fixes the direction. And reversing the order reverses the result. The magnitude of u cross v is the area of the parallelogram the two span. Half of it is the triangle's area — computable from coordinates alone. And the scalar triple product gives the parallelepiped's volume. It vanishes exactly when the three vectors are coplanar. Next up is section ten point four — Applications. Putting both products to work on lines, planes, torque and distance.

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FAQ

Questions students ask us

About The Cross Product

What is the cross product of two vectors?
The cross product u\times v is a vector perpendicular to both u and v, and it is defined only in three dimensions. Its direction follows the right-hand rule: point your right-hand fingers along u and curl them toward v, and your thumb points along u\times v. It is anticommutative, so v\times u=-(u\times v), and it is the zero vector when u and v are parallel.
How do you compute a cross product using a 3x3 determinant?
Put i, j, k in the top row, the components of u in the second row, and the components of v in the third row, then expand along the top row. The j term carries a minus sign. For u=\langle 2,3,-1\rangle and v=\langle 4,-1,2\rangle, this gives u\times v=\langle 5,-8,-14\rangle. To check the result, dot it with each input. Both 10-24+14 and 20+8-28 equal zero.
How do you find the area of a triangle in 3D using the cross product?
Take half the magnitude of the cross product of two edge vectors drawn from the same vertex. The magnitude |u\times v|=|u||v|\sin\theta equals the area of the parallelogram spanned by u and v, and the triangle is half of that. For example, if the cross product is \langle 6,-6,6\rangle, the parallelogram area is \sqrt{108}=6\sqrt{3}. The triangle's area is then 3\sqrt{3}\approx 5.196 square units.
How do you use the scalar triple product to find volume?
The volume of the parallelepiped spanned by a, b, and c is |a\cdot(b\times c)|. You can compute it as a single 3\times 3 determinant with the three vectors as its rows. The absolute value is needed because the determinant can be negative, depending on the orientation of the vectors, but a volume cannot. If the triple product equals zero, the three vectors are coplanar, which makes this the standard test for coplanarity.

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