Components of a vector | MIT 18.02SC Multivariable Calculus, Fall 2010

TL;DR

The component of 2i - 2j + k in the direction of i + j + k is 1 divided by the square root of 3. A component is the signed length of a vector's projection onto a direction vector, calculated as the dot product divided by the direction vector's length. Read on to see how the projection definition leads directly to this formula and result.

Transcript

JOEL LEWIS: Hi. Welcome back to recitation. In lecture, among other things, you've been learning about computing components of one vector in the direction of another vector. So I have a straightforward problem about that for you here. So we've got two vectors. The vector 2i minus 2j plus k. And we've got the vector i plus j plus k. And so what I'd ... Read More

Key Insights

• Component means signed projection length: The component is a scalar length associated with the projection of one vector onto another vector's direction. It is not presented merely as the projected vector itself. The sign records whether that projection follows the direction vector or points against it.
• Direction determines the denominator: In the expression (v dot w)/|w|, w is the vector specifying the direction. Its length belongs in the denominator because the dot product contains an extra factor of |w|. Dividing by |w| leaves the desired quantity |v| cos theta.
• The right triangle supplies cosine: Dropping a perpendicular from v to the line along w forms the right triangle used in the derivation. The projection lies along the adjacent direction, while v supplies the hypotenuse. Therefore, its signed length is represented by |v| cos theta.
• The dot product bridges representations: Geometry describes the component through a projection and an angle, while coordinates make it directly computable. The identity v dot w = |v||w| cos theta connects these views. This allows the angle-based expression to become a formula involving only a dot product and a vector length.
• Coordinate form simplifies evaluation: Because v and w are supplied using i, j and k coordinates, their dot product requires only products of matching entries followed by addition. For the given vectors, the calculation is 2 times 1 plus negative 2 times 1 plus 1 times 1.
• Cancellation leaves a positive dot product: The first two terms in the dot product cancel because 2 - 2 equals 0. The remaining third-coordinate product equals 1, so v dot w is 1. This positive value leads to a positive component in the stated direction.
• Vector length uses squared coordinates: The length of w = i + j + k is obtained from the square root of the sum of its squared coordinates. Since all three coordinates equal 1, the sum is 1 squared plus 1 squared plus 1 squared, giving square root of 3.
• Only two calculations are needed: Once the component formula is established, the coordinate problem reduces to finding v dot w and |w|. There is no need to calculate the angle theta separately. The dot product and direction-vector length already encode the required cosine relationship.
• The component is not |v|: The magnitude of v is the hypotenuse length in the projection diagram, not generally the requested component. Multiplication by cos theta selects the signed amount of v lying along w. The dot-product formula performs that selection without directly finding theta.
• Signs carry directional information: A projection in the same direction as w produces a positive component, while one in the opposite direction produces a negative component. The sign therefore distinguishes orientation along the chosen line, even though an ordinary geometric length alone would not.
• Scaling is corrected by |w|: The raw dot product combines the lengths of both vectors with the cosine of their angle. The requested component should retain the contribution from v but not the magnitude of the vector chosen merely to specify direction. Dividing by |w| makes that correction.
• The final scalar follows directly: Substituting the dot product 1 and direction-vector length square root of 3 into the formula yields 1 divided by the square root of 3. This is the component of 2i - 2j + k in the direction of i + j + k.

Questions & Answers

Q: How do you find the component of one vector in the direction of another?

Take the dot product of the vector being projected with the direction vector. Then divide that dot product by the length of the direction vector. This works because the dot product equals the product of both vector lengths and the cosine of their angle. Dividing by the direction vector's length leaves the signed projection length, |v| cos theta.

Q: What is the component of 2i - 2j + k in the direction of i + j + k?

The component is 1 divided by the square root of 3. The dot product is 2 times 1 plus negative 2 times 1 plus 1 times 1, which equals 1. The length of i + j + k is the square root of 1 squared plus 1 squared plus 1 squared, or square root of 3. Dividing 1 by square root of 3 gives the result.

Q: What does a vector component mean geometrically?

A component is the signed length of one vector's projection onto the direction of another vector. The projection is formed by dropping a perpendicular line to the line along the direction vector. Its length measures how much of the original vector lies along that direction. Its sign indicates whether it points with or against the chosen direction.

Q: Why is the component formula divided by the length of the direction vector?

The dot product v dot w equals |v||w| cos theta. The desired component is |v| cos theta, which does not include the factor |w|. Dividing the dot product by |w| removes that factor. The resulting expression is therefore (v dot w)/|w|.

Q: How is the dot product calculated for the given vectors?

Match the coordinates of 2i - 2j + k with those of i + j + k. Multiply each matching pair to obtain 2 times 1, negative 2 times 1 and 1 times 1. Add those products to get 2 - 2 + 1. The dot product is therefore 1.

Q: How is the length of i + j + k calculated?

Use the usual vector length formula by squaring each coordinate and adding the results. The three coordinates of i + j + k are 1, 1 and 1. Their squared sum is 1 squared plus 1 squared plus 1 squared, which equals 3. Taking the square root gives square root of 3.

Q: When is a vector component positive or negative?

The component is positive when the projection points in the same direction as the direction vector. It is negative when the projection points in the opposite direction. This sign supplements the projection's length with directional information. In the worked problem, the component is positive because the dot product equals positive 1.

Q: Do you need to calculate the angle between the vectors?

No angle calculation is needed when the vectors are provided in coordinate form. Compute their dot product and the length of the direction vector instead. The dot product formula already contains the cosine of the angle between them. Dividing the dot product by the direction vector's length directly gives the component.

Summary & Key Takeaways

• Introducing the component problem: The recitation asks for the component of the vector 2i - 2j + k in the direction of i + j + k. Because both vectors are already written in coordinate form, the problem can be solved by recalling the definition of a component and evaluating two quantities. The instructor first invites viewers to pause and attempt the calculation before presenting the solution.

• Defining projection and component: For vectors v and w, project v onto w by dropping a perpendicular line to the line containing w. The resulting vector is the projection of v onto w. The component of v in the direction of w is the length of that projection, together with a sign. It is positive when the projection points in the same direction as w and negative when it points oppositely.

• Connecting geometry to dot products: The projection diagram creates a right triangle whose hypotenuse has length |v|. If theta is the angle between v and w, the signed component is |v| cos theta. The dot product formula gives v dot w = |v||w| cos theta. Dividing that equation by |w| isolates |v| cos theta, producing the component formula (v dot w)/|w|.

• Computing the required quantities: Let v = 2i - 2j + k and w = i + j + k. Their dot product is found by multiplying corresponding coordinates and adding: 2 times 1, plus negative 2 times 1, plus 1 times 1. This simplifies to 2 - 2 + 1, so v dot w equals 1. The direction vector has length square root of 1 squared plus 1 squared plus 1 squared, which is square root of 3.

• Substituting into the formula: The component of v in the direction of w equals the dot product divided by the length of w. Substituting v dot w = 1 and |w| = square root of 3 gives 1 divided by the square root of 3. The result is positive because the computed dot product is positive, matching the definition in which a same-direction projection receives a positive sign.