Try the fastest way to create flashcards
Question

# Write $\mathbf{u}$ as the sum of a vector parallel to $\mathbf{v}$ and a vector orthogonal to $\mathbf{v}$.$\mathbf{u}=3 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}$

Solution

Verified
Step 1
1 of 3

Remember that a vector $\mathbf{u}$ can be expressed as the sum of the vector parallel to $\mathbf{v}$ and the vector orthogonal to $\mathbf{v}$ by the following expression:

$\mathbf{u}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}+\left(\mathbf{u}-\operatorname{proj}_{\mathbf{v}} \mathbf{u}\right) \tag{1}$

Where $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$ is the projection of $\mathbf{u}$ parallel to $\mathbf{v}$ and is given by:

$\operatorname{proj}_{\mathbf{v}} \mathbf{u}=\left(\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^{2}}\right) \mathbf{v}\tag{2}$

and $\left(\mathbf{u}-\operatorname{proj}_{\mathbf{v}} \mathbf{u}\right)$ is orthogonal to $\mathbf{v}$

## Recommended textbook solutions #### Thomas' Calculus

14th EditionISBN: 9780134438986 (11 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,142 solutions #### Thomas' Calculus: Early Transcendentals

11th EditionISBN: 9780321198006 (4 more)George B Thomas Jr, Joel D. Hass, Maurice D. Weir
10,574 solutions #### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (6 more)James Stewart
11,085 solutions #### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (1 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions