Proyeksi Skalar dan Proyeksi Vektor Ortogonal

Summary

This video tutorial explains the concepts of orthogonal and scalar projections, particularly focusing on vector projections. It covers how to calculate the scalar orthogonal projection of vector A onto vector B and vice versa, using formulas and step-by-step examples. The tutorial emphasizes the differences between scalar and vector projections, explaining how scalar projections determine the length of a vector, while vector projections yield a new vector result. Several example problems are solved, providing a comprehensive understanding of these concepts in vector mathematics.

Takeaways

• 😀 Scalar orthogonal projection is also called the length of an orthogonal vector projection.
• 😀 The formula for scalar orthogonal projection is calculated as the dot product of vectors divided by the magnitude of the vector in question.
• 😀 The projection of vector A onto vector B involves multiplying vector A with vector B, divided by the magnitude of vector B.
• 😀 Similarly, the projection of vector B onto vector A involves the same method, but with the magnitude of vector A in the denominator.
• 😀 When calculating the scalar orthogonal projection, you must correctly determine the order of the vectors involved.
• 😀 Example: Given vectors A = (2, -1, 3) and B = (-1, 2, -2), the scalar orthogonal projection of A onto B involves specific multiplication and division steps.
• 😀 Scalar orthogonal projection yields a scalar value, representing the length of the projection along a certain direction.
• 😀 The formula for vector orthogonal projection uses the same approach as scalar projection but results in a vector instead of a scalar.
• 😀 For vector projections, the formula involves multiplying the scalar projection result by the vector being projected onto.
• 😀 The concept of orthogonal projections is used in various fields such as mathematics and physics for decomposing vectors into components along specific directions.

Q & A

What is scalar orthogonal projection?

Scalar orthogonal projection refers to the projection of a vector onto another vector, where the result is a scalar representing the magnitude of the projection. It is calculated by dividing the dot product of the two vectors by the magnitude of the second vector.

What formula is used to calculate the scalar orthogonal projection of vector 'a' onto vector 'b'?

The formula for the scalar orthogonal projection of vector 'a' onto vector 'b' is: P_a on b = (a · b) / |b|, where a · b is the dot product of the vectors and |b| is the magnitude of vector 'b'.

How is the orthogonal vector projection different from scalar projection?

The orthogonal vector projection results in a vector, not a scalar. The formula involves multiplying the scalar projection by the vector 'b', resulting in a vector that represents the projection in the direction of 'b'.

What formula is used to compute the orthogonal vector projection?

The formula for the orthogonal vector projection of vector 'a' onto vector 'b' is: P_a on b = ((a · b) / |b|^2) * b, where a · b is the dot product of the vectors, and |b|^2 is the square of the magnitude of vector 'b'.

What does the result of an orthogonal vector projection represent?

The result of an orthogonal vector projection represents the vector that lies in the direction of vector 'b' and has the same magnitude as the projection of vector 'a' onto 'b'. It is the closest vector to 'a' that lies along the line defined by 'b'.

How do you compute the dot product of two vectors?

To compute the dot product of two vectors, multiply their corresponding components and then sum the results. For vectors a = (x1, y1, z1) and b = (x2, y2, z2), the dot product is: a · b = x1 * x2 + y1 * y2 + z1 * z2.

What is the formula for calculating the magnitude of a vector?

The magnitude of a vector is calculated as the square root of the sum of the squares of its components. For a vector a = (x, y, z), the magnitude is: |a| = √(x² + y² + z²).

In the example, how do you calculate the scalar orthogonal projection of vector 'a' onto vector 'b'?

For vectors a = (2, -1, 3) and b = (-1, 2, -2), the scalar orthogonal projection is computed using the formula: P_a on b = (a · b) / |b|. First, calculate the dot product a · b = (2 * -1) + (-1 * 2) + (3 * -2) = -2 - 2 - 6 = -10. Then, calculate the magnitude of b: |b| = √((-1)² + 2² + (-2)²) = √1 + 4 + 4 = √9 = 3. The scalar projection is then: P_a on b = -10 / 3.

What steps are involved in calculating the vector orthogonal projection of 'a' onto 'b'?

To calculate the vector orthogonal projection of 'a' onto 'b', first calculate the scalar projection as described in the previous question. Then, multiply the scalar projection by the vector 'b'. For the example given, after calculating the scalar projection, multiply it by the vector 'b' to get the final vector projection.

What is the significance of rationalizing the denominator when calculating projections?

Rationalizing the denominator helps to simplify the final expression. For instance, when the denominator involves a square root, rationalizing eliminates the square root in the denominator, making the result easier to interpret and work with.