2. Vector Projection: The second step is to multiply this scalar projection by the unit vector in the direction of b. The unit vector of b is given by \( \mathbf{b} / ||\mathbf{b}|| \), so the vector projection of a onto b is: \( \text{proj}_{\mathbf{b}}(\mathbf{a}) = c \times \left(\frac{\mathbf{b}}{||\mathbf{b}||}\right) = \left(\frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{b}||}\right) \times \left(\frac{\mathbf{b}}{||\mathbf{b}||}\right) = \left(\frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{b}||^2}\right) \mathbf{b} \) This gives a new vector that points in the direction of b and has length equal to the scalar projection of a onto b.
a = (3, -2, 5)
b = (1, 4, -2)
Step 1: Calculate the dot product of a and b.
The dot product of two vectors a and b is given by:
\( \mathbf{a} \cdot \mathbf{b} = a_x \cdot b_x + a_y \cdot b_y + a_z \cdot b_z \)
Substitute the values:
\( \mathbf{a} \cdot \mathbf{b} = (3 \cdot 1) + (-2 \cdot 4) + (5 \cdot -2) = 3 – 8 – 10 = -15 \)
Step 2: Calculate the magnitude of b.
The magnitude of a vector b is given by:
\( ||\mathbf{b}|| = \sqrt{b_x^2 + b_y^2 + b_z^2} \)
Substitute the values:
\( ||\mathbf{b}|| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \)
Step 3: Calculate the scalar projection of a onto b.
The scalar projection of a onto b is given by:
\( c = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{b}||} \)
Substitute the values:
\( c = \frac{-15}{\sqrt{21}} \)
Step 4: Calculate the vector projection of a onto b.
The vector projection of a onto b is given by:
\( \text{proj}_{\mathbf{b}}(\mathbf{a}) = c \times \left(\frac{\mathbf{b}}{||\mathbf{b}||}\right) \)
Substitute the values and multiply:
\( \text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{-15}{\sqrt{21}} \times \left(\frac{1}{\sqrt{21}}, \frac{4}{\sqrt{21}}, \frac{-2}{\sqrt{21}}\right) \)
Simplify:
\( \text{proj}_{\mathbf{b}}(\mathbf{a}) = \left(\frac{-15}{21}, \frac{-60}{21}, \frac{30}{21}\right) \)
Step 5: Simplify the result.
The simplified vector projection is:
\( \text{proj}_{\mathbf{b}}(\mathbf{a}) = \left(\frac{-5}{7}, \frac{-20}{7}, \frac{10}{7}\right) \)
So, the projection of vector a onto vector b in 3D is \( \left(\frac{-5}{7}, \frac{-20}{7}, \frac{10}{7}\right) \). This means that the component of a in the direction of b is \( \frac{-5}{7} \) in the x-direction, \( \frac{-20}{7} \) in the y-direction, and \( \frac{10}{7} \) in the z-direction.
u = (2, 1, 3)
v = (3, -2, 4)
Step 1: Calculate the dot product of u and v.
The dot product of two vectors u and v is given by:
\( \mathbf{u} \cdot \mathbf{v} = u_x \cdot v_x + u_y \cdot v_y + u_z \cdot v_z \)
Substitute the values:
\( \mathbf{u} \cdot \mathbf{v} = (2 \cdot 3) + (1 \cdot -2) + (3 \cdot 4) = 6 – 2 + 12 = 16 \)
Step 2: Calculate the magnitude of v.
The magnitude of a vector v is given by:
\( ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} \)
Substitute the values:
\( ||\mathbf{v}|| = \sqrt{3^2 + (-2)^2 + 4^2} = \sqrt{9 + 4 + 16} = \sqrt{29} \)
Step 3: Calculate the scalar projection of u onto v.
The scalar projection of u onto v is given by:
\( c = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||} \)
Substitute the values:
\( c = \frac{16}{\sqrt{29}} \)
Step 4: Calculate the vector projection of u onto v.
The vector projection of u onto v is given by:
\( \text{proj}_{\mathbf{v}}(\mathbf{u}) = c \times \left(\frac{\mathbf{v}}{||\mathbf{v}||}\right) \)
Substitute the values and multiply:
\( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{16}{\sqrt{29}} \times \left(\frac{3}{\sqrt{29}}, \frac{-2}{\sqrt{29}}, \frac{4}{\sqrt{29}}\right) \)
Simplify:
\( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \left(\frac{48}{29}, \frac{-32}{29}, \frac{64}{29}\right) \)
Step 5: Simplify the result.
The simplified vector projection is:
\( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \left(\frac{48}{29}, \frac{-32}{29}, \frac{64}{29}\right) \)
So, the projection of vector u onto vector v in 3D is \( \left(\frac{48}{29}, \frac{-32}{29}, \frac{64}{29}\right) \). This means that the component of u in the direction of v is \( \frac{48}{29} \) in the x-direction, \( \frac{-32}{29} \) in the y-direction, and \( \frac{64}{29} \) in the z-direction.
Imagine a tiny person walking along vectors in 3D space to understand the concept of vector projection.
Imagine the tiny person starting at the origin (0, 0, 0) and standing on vector u = (2, 1, 3). This vector represents the direction the person is facing and the distance they move in each step.
Now, let’s introduce another vector v = (3, -2, 4). This vector represents the direction of a path or a direction the person wants to walk towards.
Step 1: The dot product of u and v (16) tells us how much the tiny person is aligned with the direction of vector v. If the person is walking exactly in the direction of v, the dot product will be the magnitude of v (which is \( \sqrt{29} \)).
Step 2: The magnitude of v (\( \sqrt{29} \)) gives the length of the path the tiny person wants to walk on. It represents how far the person wants to walk along the direction of v.
Step 3: The scalar projection (\( c = \frac{16}{\sqrt{29}} \)) tells us how far the tiny person has actually moved in the direction of v. If the scalar projection is positive, the person has moved in the same direction as v. If it’s negative, the person has moved in the opposite direction of v.
Step 4: The vector projection of u onto v (\( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \left(\frac{48}{29}, \frac{-32}{29}, \frac{64}{29}\right) \)) gives the actual displacement of the tiny person along the path represented by v. It tells us how far the person has moved along v from the starting point (0, 0, 0).
So, if the tiny person is walking along the vector u = (2, 1, 3) and wants to move along the direction represented by v = (3, -2, 4), the vector projection \(\left(\frac{48}{29}, \frac{-32}{29}, \frac{64}{29}\right)\) tells us the displacement of the person along the path v.
In simple terms, the vector projection shows the tiny person’s progress along the path or direction represented by vector v while starting from the origin and walking along vector u. It allows us to understand how much the person is moving in the desired direction and how far they have gone along that direction.